On cyclic codes and quasi-cyclic codes over Zq+ uZq
arXiv:1501.03924v3 [cs.IT] 24 Jan 2015
On cyclic codes and quasi-cyclic codes over Zq + uZq Jian Gao∗1 , Fang-Wei Fu1 , Ling Xiao2 , Rama Krishna Bandi3 1. Chern Institute of Mathematics and LPMC, Nankai University Tianjin, 300071, P. R. China 2. School of Science, Shandong University of Technology Zibo, 255091, P. R. China 3. Department of Mathematics,Indian Institute of Technology Roorkee Roorkee, 247667, India E-mail: [email protected]
Abstract Let R = Zq + uZq , where q = ps and u2 = 0. In this paper, some structural properties of cyclic codes and quasi-cyclic (QC) codes over the ring R are considered. A QC code of length ℓn with index ℓ over R is viewed both as in the conventional row circulant form and also as an R[x]/(xn − 1)-submodule of GR(R, ℓ)[x]/(xn − 1), where GR(R, ℓ) is the Galois extension ring of degree ℓ over R. A necessary and sufficient condition for cyclic codes over the ring R to be free is obtained and a BCH-type bound for them is also given. A sufficient condition for 1-generator QC codes to be R-free is given. Some distance bounds for 1-generator QC codes are also discussed. The decomposition and dual of QC codes over R are also briefly discussed. Keywords Cyclic codes; Quasi-cyclic codes; Hensel lift; 1-Generator quasi-cyclic codes
Mathematics Subject Classification (2000) 11T71 · 94B05 · 94B15
1
Introduction
Quasi-cyclic (QC) codes are an important class of linear codes and have some good algebra structures [3]-[10]. Recently, there are some research papers about QC codes over finite chain rings [1, 3, 4, 5, 6, 12]. Aydin et al. studied QC codes over Z4 and obtained some new binary codes using the usual Gray map [1]. Moreover, they characterized cyclic codes corresponding to free modules in terms of their generator polynomials. Bhaintwal et al. discussed QC codes over the prime integer residue ring Zq [3]. They viewed a QC code of length mℓ with index ℓ as an Zq [x]/(xm − 1)-submodule of GR(q, ℓ)[x]/(xm − 1), where GR(q, ℓ) was the ℓth Galois extension ring of Zq . A sufficient condition for 1-generator QC code to be Zq -free was given and some distance bounds for 1-generator QC codes were also discussed. Cui et al. considered 1-generator QC codes over Z4 [4]. Under some conditions,
1
they gave the enumeration of quaternary 1-generator QC codes of length mℓ with index ℓ, and described an algorithm to obtain one and only one generator for each 1-generator QC code. Based on the idea in [4], 1-generator QC codes over another special finite chain ring Fq + uFq were considered [5], and then were generalized to the general finite chain rings [6]. However, it should be noted that all research papers discussed above base on one fact that the block length of the QC code is coprime with the characteristic of finite chain rings. Siap, et al. studied 1-generator QC codes of arbitrary lengths over finite chain ring F2 + uF2 [12]. They gave the generating set and the free condition of the 1-generator QC code. Using Gray map, they also got some optimal binary linear codes over finite field F2 . In these studies, the group rings associated with QC codes are finite chain rings. Recently, Zhu et al. considered linear codes over the finite non-chain ring Fq + vFq [15, 16]. In [16], they studied the cyclic codes over F2 + vF2 . It has shown that cyclic codes over this ring are principally generated. In the subsequent paper [15], they investigated a class of constacyclic codes over Fp + vFp . In that paper, the authors proved that the image of a (1 − 2v)-constacyclic code of length n over Fp + vFp under the Gray map is a cyclic code of length 2n over Fp . Furthermore, they also asserted that (1 − 2v)-constacyclic codes over Fp + vFp are also principally generated. More recently, Yildiz and Karadeniz [14] studied the linear codes over the non-principal ring Z4 + uZ4 , where u2 = 0. They introduced the MacWilliams identities for the complete, symmetrized and Lee weight enumerators. They also gave three methods to construct formally self-dual codes over Z4 + uZ4 . Bandi and Bhaintwal studied some structural properties of cyclic codes of odd length over Z4 + uZ4 , where u2 = 0 [2]. They provided the general form of the generators of a cyclic code over Z4 + uZ4 , and they also determined a necessary condition and a sufficient condition for cyclic codes of odd length over Z4 + uZ4 to be (Z4 + uZ4 )-free. Let R = Zq + uZq , where p is some prime, q = ps and u2 = 0. It is natural to ask if we can also study some structural properties of cyclic codes and QC codes over R. In this paper, we mainly consider this issue. The paper is organized as follows. In Section 2, we introduce some basic results on the ring R. In Section 3, we study the cyclic codes over R. We determine the generator from of cyclic codes. And, we also give a necessary and sufficient condition for cyclic codes over R to be R-free. In Section 4, we discuss module structures of QC codes over R, which studied by the same method of QC codes over finite fields [8] and Galois rings [3]. This point of view for studying QC codes could give a lower bound on the minimum Hamming distance and a construct method of linear codes over Zq . In Section 5, 1-generator QC codes over R are studied. A necessary and sufficient condition for QC cods over R to be R-free is given. Further, we also give a minimum Hamming distance bound on 1-generator QC codes. In Section 6, the decomposition and dual of QC codes are discussed briefly.
2
2
The ring Zq + uZq
Let R = Zq + uZq , where p is some prime, q = ps and u2 = 0. If q = 4, then R = Z4 + uZ4 . The ring R is isomorphic to the quotient ring Zq [u]/(u2 ) and R = {a + bu : a, b ∈ Zq }. Further, The ideals of R are of the following forms: (i) (pi ) for 0 ≤ i ≤ s; (ii) (pk u) for 0 ≤ k ≤ s − 1; (iii) (pj + u) for 1 ≤ j ≤ s − 1; (iv) (pj , u) for 1 ≤ j ≤ s − 1. Therefore, there are 4s − 1 ideals of R. For example, there are 4 × 2 − 1 = 7 ideals of Z4 + uZ4 . R is a local ring with the characteristic q and the maximal ideal (p, u). But it is not a chain ring, since both of the ideals (p) and (u) are not included each other. Further, R is not principal since the ideal (p, u) can not be generated by any single element of this ideal. Define a map −
: R → R/(p, u)
(1)
r = a + bu 7→ a (modp)
The map − is a ring homomorphism and R/(p, u) is denoted by the residue field R. Since for any r = a + bu, a ∈ Zq then R is isomorphic to the finite field Fp . Let R[x] be the polynomial ring over R. The map − can be extended to R[x] to R[x] in the usual way. The image of any element f (x) ∈ R[x] under this map is denoted by f (x). Two polynomials f (x) and g(x) are said to be coprime over R if and only if there are two polynomials a(x) and b(x) in R[x] such that a(x)f (x) + b(x)g(x) = 1.
(2)
A polynomial f (x) ∈ R[x] is said to be basic irreducible if f (x) is irreducible in R[x] and basic primitive if f (x) is primitive in R[x]. In the following, we consider the factorization of xn − 1 over R. We assume that gcd(n, q) = 1 throughout this paper. r
Lemma 1. Let g(x) be a irreducible polynomial over Fp , where g(x)|(xp −1 − 1) for some positive integer r. Then there exists a unique basic irreducible polynomial f (x) such that r f (x) = g(x) and f (x)|(xp −1 − 1) over R. r
Proof. Let xp −1 − 1 = g(x)m(x). Since gcd(n, q) = 1, it follows that g(x) has no multiple roots. Clearly, x ∤ g(x). Then g(x) has the unique Hensel lift f (x) over Zq such that r f (x)|(xp −1 − 1) (see Theorem 13.10 in [13]). Since Zq is a subring of R, it follows that r r the factorization of xp −1 − 1 is still valid over R. It means that f (x)|(xp −1 − 1) over R. Further, g(x) is irreducible over Fp deduces f (x) is basic irreducible over R. 3
We call the polynomial f (x) in Lemma 1 the Hensel lift of g(x) to R[x]. Since gcd(n, q) = 1, it follows that the polynomial xn − 1 can be factored uniquely into pairwise coprime basic irreducible polynomials over R, i.e. xn − 1 = f1 (x)f2 (x) · · · ft (x),
(3)
where, for l = 1, 2, . . . , t, fl (x) is a basic irreducible polynomial over R. m Let T = {0, 1, ξ, . . . , ξ p −2 } be the Teichm¨ uller set of Galois ring GR(q, m), where GR(q, m) is the mth Galois extension ring of Zq and ξ is a basic primitive element of GR(q, m). Then for each a ∈ GR(q, m), it can be written as a = a0 + a1 p + · · · + as−1 ps−1 where a0 , a1 , . . . , as−1 ∈ T . This is called the p-adic representation of the element of GR(q, m) (see the Section 3 of Chapter 14 in [13]). Now we consider the Galois extension of R. Let f (x) be a basic irreducible polynomial of degree m over R. The the mth Galois extension R[x]/(f (x)) is denoted by GR(R, m). Let α be a root of f (x). Then 1, α, . . . , αm−1 form a set of R-free basis and GR(R, m) = {r0 + r1 α + · · · + rm−1 αm−1 : r0 , r1 , . . . , rm−1 ∈ R}.
(4)
The ring GR(R, m) is a local ring with maximal ideal ((p, u) + (f (x))). Its residue field is isomorphic to Fpm . Moreover, GR(R, m) ∼ = GR(q, m)[u]/(u2 ) ∼ = GR(q, m) + uGR(q, m).
(5)
Therefore, for any element r = a + bu ∈ GR(q, m), r=
s−1 X
ai p i + u
Ps−1 i=0
ai p i , b =
b i pi ,
(6)
i=0
i=0
where ai , bi ∈ T and a =
s−1 X
Ps−1 i=0
b i pi .
Ps−1 Ps−1 Lemma 2. For any r = i=0 ai pi + u i=0 bi pi ∈ GR(R, m), r is a unit under multiplication if and only if a0 6= 0. Proof. One can verify that r is a unit under multiplication of R if and only if rk is a unit under multiplication of R for any positive integer k. Particularly, this is valid for k = q. Note that, for any r ∈ R, rq = aq0 ∈ T . Therefore r is a unit if and only if aq0 is a unit if and only if aq0 6= 0 if and only if a0 6= 0. From Lemma 2, we have that the group of units of GR(R, m) denoted by GR(R, m)∗ is given by s−1 s−1 X X bi pi : ai , bi ∈ T , a0 6= 0}. (7) GR(R, m)∗ = { ai p i + u s=0
i=0
Let m
GC = {1, ξ, . . . , ξ p 4
−2
}
and GA = {1 +
s−1 X
j
aj p + u
s−1 X
bi pi : aj , bi ∈ T }.
i=0
j=1
Theorem 1. GR(R, m)∗ = GC × GA , and |GR(R, m)∗ | = (pm − 1) · (p2s−1 )m . m
Proof. Let GC = {1, ξ, . . . , ξ p −2 }. Then GC is a multiplicative cyclic group of order Ps−1 Ps−1 i Ps−1 Ps−1 i pm − 1. Let r = i=0 ai pi + u i=0 bi p be a unit of R, i.e. r = i=0 ai pi + u i=0 bi p ∈ GR(R, m)∗ . Define a group homomorphism as follows Γ : GR(R, m)∗ → GC r=
s−1 X
ai p i + u
s−1 X
bi pi 7→ a0 .
i=0
i=0
Clearly, Γ is a surjective map. Then, by the first homomorphic theorem, we have GR(R, m)∗ /KerΓ ∼ = GC . Clearly, KerΓ = {1 +
s−1 X j=1
j
aj p +
s−1 X
bi pi : a1 , a2 , . . . , as−1 , b0 , b1 , . . . , bs−1 ∈ T }.
i=0
Denote KerΓ by GA . Then GR(R, m)∗ /GA ∼ = GC and |GR(R, m)∗ | = |GC | · |GA | = (pm − 2s−1 m 1) · (p ) The set of all zero divisors of GR(R, m) is {
s−1 X
aj p j + u
s−1 X
bi pi },
(8)
i=0
j=1
which is the maximal ideal of GR(R, m). Lemma 3. Let f (x) and g(x) be polynomials of R[x]. Then f (x) and g(x) are coprime if and only if f (x) and g(x) are coprime over R = Fp . Proof. If f (x) and g(x) are coprime over R, then there are polynomials a(x), b(x) ∈ R[x] such that a(x)f (x) + b(x)g(x) = 1, which implies that a(x)f (x) + b(x)g = 1 with a(x), b(x), f (x), g(x) ∈ Fp [x]. Therefore f (x) and g(x) are coprime over Fp .
5
On the other hand, if f (x) and g(x) are coprime over Fp , then there are polynomials a(x) and b(x) ∈ Fp [x] such that a(x)f (x) + b(x)g = 1, which implies that a(x)f (x) + b(x)g(x) = 1 + pr(x) + ut(x) for some p(x), t(x) ∈ R[x]. Let λ(x) =
s−1 X
(−pr(x))i and τ (x) = 1 − ut(x)λ(x).
i=0
Let κ(x) = λ(x)τ (x). Then κ(x)a(x)f (x) + κ(x)b(x)g(x) = 1, which implies that f (x) and g(x) are coprime over R. Theorem 2. Let GR(R, m) = R[x]/(f (x)) be the mth Galois extension of R, where f (x) is a basic irreducible polynomial with degree m over R. Then the ideals of GR(R, m) are precisely (i) (pi + (f (x))) for 0 ≤ i ≤ s; (ii) (pk u + (f (x))) for 0 ≤ k ≤ s − 1; (iii) (pj + u + (f (x))) for 1 ≤ j ≤ s − 1; (iv) ((pj , u) + (f (x))) for 1 ≤ j ≤ s − 1. Proof. Let I be an ideal of R. If I is zero, then I = (ps + (f (x))) of R[x]/(f (x)). In the following, we determine the nonzero ideals of R[x]/(f (x)). Let g(x) ∈ I. Since f (x) is basic irreducible over R, it follows that f (x) is irreducible over Fp . Therefore gcd(f (x), g(x)) = 1 or f (x). If gcd(f (x), g(x)) = 1, then gcd(f (x), g(x)) = 1 which implies that there are polynomials a(x), b(x) ∈ R[x] such that a(x)f (x) + bg(x) = 1. It means that g(x) is a unit of R[x]/(f (x)), i.e. I = R[x]/(f (x)). If gcd(f (x), g(x)) = f (x), then there are polynomials a(x), b(x), c(x) ∈ R[x] such that g(x) = a(x)f (x) + pb(x) + uc(x). Therefore g(x) ∈ ((p, u) + (f (x))) of R[x]/(f (x)). Since the ideals contained in g(x) ∈ ((p, u) + (f (x))) are as the form in this theorem. The result follows.
3
Cyclic codes over R
Let Rn be a free R-module of rank n, i.e. Rn = {(c0 , c1 , . . . , cn−1 ) : c0 , c1 , . . . , cn−1 ∈ R}. Let C be a nonempty set of Rn . C is called a linear code of length n if and only if C is an R-submodule of Rn . Let T be the cyclic shift operator. If for any c = (c0 , c1 , . . . , cn−1 ) ∈ C
6
the T (c) = (cn−1 , c0 , . . . , cn−2 ) is also in C , we say C is a cyclic code of length n over R. Define an R-module isomorphism as follows Φ : Rn → R[x]/(xn − 1) (c0 , c1 , . . . , cn−1 ) 7→ c0 + c1 x + · · · + cn−1 xn−1 . One can verify that C is a cyclic code of length n over R if and only if Φ(C ) is an ideal of the quotient ring R[x]/(xn − 1). Sometimes, we identity the cyclic code of length n over R with the ideal of R[x]/(xn − 1). Review that xn − 1 = f1 (x)f2 (x) · · · ft (x), where, for each l = 1, 2, . . . , t, fl (x) is a n −1 by fbl (x). Since fl (x) and fbl (x) are basic irreducible polynomial over R. Denote xfl (x) coprime to each other, it follows that there are polynomials al (x), bl (x) in R[x] such that al (x)fl (x) + bl (x)fbl (x) = 1. Let el (x) = bl (x)fbl (x) + (xn − 1) and Ri = el (x)R[x]/(xn − 1). Then we have R[x]/(xn − 1) = R1 ⊕ R1 ⊕ · · · ⊕ Rt . (9) For any l = 1, 2, . . . , t, the map Ψ : R[x]/(fl (x)) → Rl k(x) + (fl (x)) 7→ (k(x) + (xn − 1))el (x)
(10)
is an isomorphism of rings. Therefore, R[x]/(xn − 1) ∼ = R[x]/(f1 (x)) × R[x]/(f2 (x)) × · · · × R[x]/(ft (x)).
(11)
Lemma 4. Let xn − 1 = f1 (x)f2 (x) · · · ft (x) where, for each l = 1, 2, . . . , t, fl (x) is a basic irreducible polynomial over R. Then under the map Ψ, the ideals of R[x]/(fl (x)) are mapped into (i) (pi fbl (x) + (xn − 1) for 0 ≤ i ≤ s; (ii) (pk ufbl (x) + (xn − 1)) for 0 ≤ k ≤ s − 1; (iii) ((pj + u)fbl (x) + (xn − 1)) for 1 ≤ j ≤ s − 1; (iv) ((pj , u)fbl (x) + (xn − 1)) for 1 ≤ j ≤ s − 1 of Rl . Proof. Under the ring isomorphism Ψ, we have 1 + (fl (x)) 7→ (1 + (xn − 1))el . Since el = bl (x)fbl (x) + (xn − 1), it follows that 1 + (fl (x)) 7→ bl (x)fbl (x) + (xn − 1). Clearly, bl (x)fbl (x) + (xn − 1) ∈ (fbl (x) + (xn − 1)). 7
Multiplying both sides of al (x)fl (x) + bl (x)fbl (x) = 1 by fbl (x), we obtain bl (x)fbl (x)fbl (x) + al (x)(xn − 1) = fbl (x). Then bl (x)fbl (x)fbl (x) + (xn − 1) = fbl (x) + (xn − 1),
which implies that fbl (x) + (xn − 1) ∈ (bl (x)fbl (x) + (xn − 1)). Therefore, (bl (x)fbl (x) + (xn − 1)) = (fbl (x) + (xn − 1)) and the image of (1 + (fi (x))) under the ring isomorphism Ψ is (fbl (x) + (xn − 1)). The remainder cases can also be verified in the same way. By Theorem 2, Eq. (11) and Lemma 4, we have the following result directly. Theorem 3. Let xn − 1 = f1 (x)f2 (x) · · · ft (x) where, for each l = 1, 2, . . . , t, fl (x) is a basic irreducible polynomial over R. Then there are (4s − 1)t cyclic codes of length n over R. Further, any cyclic code is the sum of the ideals of Rl . Example 1. Consider a cyclic code of length 3 over Z4 + uZ4 . Since x3 − 1 = (x − 1)(x2 + x + 1), it follows that there are (4 × 2 − 1)2 = 72 = 49 cyclic codes of length 3 over Z4 + uZ4 . Let f1 = x − 1 and f2 = x2 + x + 1. In the following Table 1, we list all of cyclic codes of length 3 over Z4 + uZ4 . Table 1: All cyclic codes of length 3 over Z4 + uZ4 0 (f1 ) (2f1 ) (uf1 ) (2uf1 ) ((2 + u)f1 ) ((2, u)f2 ) ((2 + u)f2 ) (2 + u) ((2, u)f1 , uf2 )
(f2 ) (1) (2f1 , f2 ) (uf1 , f2 ) (2uf1 , f2 ) ((2 + u)f1 , f2 ) (f1 , (2, u)f2 ) (f1 , (2 + u)f2 ) ((2 + u)f1 , (2, u)f2 ) ((2, u)f1 , 2uf2 )
(2f2 ) (f1 , 2f2 ) (2) (uf1 , 2f2 ) (2uf1 , 2f2 ) ((2 + u)f1 , 2f2 ) (2f1 , (2, u)f2 ) (2uf1 , (2 + u)f2 ) ((2, u)f1 ) ((2, u)f1 , (2 + u)f2 )
(uf2 ) (f1 , uf2 ) (2f1 , uf2 ) (u) (2uf1 , uf2 ) ((2 + u)f1 , uf2 ) (uf1 , (2, u)f2 ) (2f1 , (2 + u)f2 ) ((2, u)f1 , f2 ) (2, u)
(2uf2 ) (f1 , 2uf2 ) (2f1 , 2uf2 ) (uf1 , 2uf2 ) (2u) ((2 + u)f1 , 2uf2 ) (2uf1 , (2, u)f2 ) (uf1 , (2 + u)f2 ) ((2, u)f1 , 2f2 )
In fact, the cyclic code C also has the following general form of generators. Theorem 4. Let C be a cyclic code of length n over R. Then C = (f0 (x) + uf1 (x), ug1 (x)) with f0 (x), f1 (x), g1 (x) ∈ Zq [x] and g1 (x)|f0 (x). Proof. Define a surjective homomorphism from R to Zq as ψ(a + bu) = a for any a + bu ∈ R. Extend ψ to the polynomial ring R[x]/(xn − 1) as ψ(a0 + a1 + · · · + an−1 xn−1 ) = ψ(a0 ) +
8
ψ(a1 )x + · · · + ψ(an−1 )xn−1 for any polynomial a0 + a1 x + · · · + an−1 ∈ R[x]/(xn − 1). Let C be a cyclic code of length n over R, and restrict ψ to C . Define a set J = {f (x) ∈ Zq [x]/(xn − 1) : uf (x) ∈ Kerψ}. Clearly, J is an ideal of Zq [x]/(xn −1), which implies that there is a polynomial g1 (x) ∈ Zq [x] such that J = (g1 (x)). It means that Kerψ = (ug1 (x)). Further, the image of C under the map ψ is also an ideal of Zq [x]/(xn − 1). Then there is a polynomial f0 (x) ∈ Zq [x] such that ψ(C ) = (f0 (x)). Hence C = (f0 (x) + uf1 (x), ug1 (x)). Clearly, u(f0 (x) + uf1 (x)) = uf0 (x) ∈ Kerψ implying g0 (x)|f0 (x). Lemma 5. Let C = (f0 (x) + uf1 (x), ug1 (x)) be a cyclic code of length n over R. If g1 (x) is a monic polynomial over R, then we can assume that deg(f1 (x)) < deg(g1 (x)). Proof. If deg(g1 (x)) ≤ deg(f1 (x)), then there are polynomials m(x), r(x) ∈ R[x] such that f1 (x) = g1 (x)m(x) + r(x) with deg(r(x)) < deg(g1 (x)) or r(x) = 0. Then C = (f0 (x) + ug1 (x)m(x) + ur(x), ug1 (x)). Let C1 = (f0 (x) + ur(x), ug1 (x)). Clearly, C ⊆ C1 . Further, f0 (x) + ur(x) = f0 (x) + ug1 (x)m(x) + ur(x) + u(q − 1)g1 (x)m(x), it follows that C1 ⊆ C . Thus C = C1 . Lemma 6. Let C = (f0 (x) + uf1 (x), ug1 (x)) be a cyclic code of length n over R. If f0 (x) = g1 (x), then C = (f0 (x) + uf1 (x)). Further, if g1 (x) is monic over R, then (f0 (x) + uf1 (x))|(xn − 1). Proof. Clearly, (f0 (x) + uf1 (x)) ⊆ C . Further, since u(f0 (x) + uf1 (x)) = uf0 (x) = ug1 (x), it follows that C ⊆ (f0 (x) + uf1 (x)). Thus C = (f0 (x) + uf1 (x)). Since f0 (x) = g1 (x) and g1 is monic, then f0 (x) + uf1 (x) is also monic over R by Lemma 5. Therefore there are polynomials a(x), b(x) ∈ R[x] such that xn − 1 = a(x)(f0 (x) + uf1 (x)) + b(x) with b(x) = 0 or deg(b(x)) < deg(f0 (x)). Since b(x) ∈ C , it follows that b(x) = 0. Thus (f0 (x) + uf1 (x))|(xn − 1) over R. Theorem 5. Let C = (f0 (x) + uf1 (x), ug1 (x)) be a cyclic code of length n over R and f0 (x), g1 (x) be monic over Zq . Let deg(f0 (x)) = k0 and deg(g1 (x)) = k1 . Then the set β = {(f0 (x) + uf1 (x)), . . . , xn−k0 −1 (f0 (x) + uf1 (x)), ug1 (x), . . . , xk0 −k1 −1 ug1 (x)} forms the minimum generating set of C , and |C | = q 2n−k0 −k1 . 9
Proof. Let γ = {(f0 (x) + uf1 (x)), . . . , xn−k0 −1 (f0 (x) + uf1 (x)), ug1 (x), . . . , xn−k1 −1 ug1 (x)}. Then γ spans the cyclic code C . Further, it is sufficient to show that β spans γ, which follows that β also spans C . Now we only need to show that β is linearly independent. For simplicity, we denote f (x) as f0 (x) + uf1 (x). Since f0 is monic, then the constant coefficient of f (x) is a unit of R by the fact that f0 (x) is the generator polynomial of some cyclic code of length n over Zq . Let a0 f (x) + a1 xf (x) + · · · + an−k0 −1 xn−k0 −1 f (x) = 0.
(12)
Let F0 be the constatant coefficient of f (x). Then a0 F0 = 0, which implies that a0 = 0. Therefore the Eq. (12) becomes a1 f (x)+a2 xf (x)+· · ·+an−k0 −1 xn−k0 −2 f (x) = 0. Similarly, a1 = a2 = · · · = an−k0 −1 = 0. Thus (f0 (x)+uf1 (x)), x(f0 (x)+uf1 (x)), . . . , xn−k0 −1 (f0 (x)+ uf1 (x)) are R-linear independent. One can also prove that ug1 (x), uxg1 (x), . . . , uxn−k1 −1 g1 (x) are Zq -linear independent. Thus |C | = q 2n−k0 −k1 . We now consider the cyclic code C as a principal ideal of R[x]/(xn − 1). Theorem 6. Let C be a principal generated cyclic code of length n over R. Then C is free if and only if there is a monic polynomial g(x) such that C = (g(x)) and g(x)|(xn − 1). Moreover, the set {g(x), xg(x), . . . , xn−deg(g(x))−1 g(x)} forms the minimum generating set of C and |C | = q 2(n−deg(g(x))) . Proof. Suppose that C = (g(x)) is R-free. Then the set {g(x), xg(x), . . . , xn−deg(g(x))−1 g(x)} form the R-basis of C . Since xn−degg(x) ∈ C , it follows that xn−deg(g(x)) can be written as a linear combination of the elements g(x), xg(x), . . . , xn−deg(g(x))−1 g(x), i.e. xn−deg(g(x)) g(x)+ Pn−deg(g(x))−1 P ai xi g(x) = 0. Let a(x) = n−deg(g(x))−1 ai xi +xn−deg(g(x)) . Then a(x)g(x) = i=0 i=0 0 in R[x]/(xn − 1), which implies that (xn − 1)|g(x)a(x). Since g(x)a(x) is monic and deg(g(x)a(x)) = n, it follows that xn − 1 = g(x)a(x) implying g(x)|(xn − 1). On the other hand, if C = (g(x)) with g(x)|(xn −1). Then g(x), xg(x), . . . , xn−deg(g(x))−1 g(x) span C . In the following, we prove that g(x), xg(x), . . . , xn−deg(g(x))−1 g(x) are R-linear independent. Let g(x) = g0 + g1 x + · · · + xdeg(g(x)) and a0 g(x) + a1 xg(x) + · · · + an−deg(g(x))−1 xn−deg(g(x))−1 g(x) = 0.
(13)
Since g0 is a unit of R and a0 g0 = 0, it follows that a0 = 0. Then the Eq. (13) becomes a1 g(x) + a2 xg(x) + · · · + an−deg(g(x))−1 xn−deg(g(x))−2 g(x) = 0. Similarly, we have a1 = a2 = · · · = an−deg(g(x))−1 = 0. Thus g(x), xg(x), . . . , xn−deg(g(x))−1 g(x) are R-linear independent, i.e. C is free over R.
10
Theorem 7. (BCH-type Bound) Let C = (g(x)) be a free cyclic code of length n over R. Suppose that g(x) has roots ξ b , ξ b+1 , . . . , ξ b+δ−2 , where ξ is a basic primitive nth root of unity in some Galois extension ring of R. Then the minimum Hamming distance of C is dH (C ) ≥ δ. Proof. The proof process is similar to that of Proposition 2 in [3]. Example 2. Let R = Z8 + uZ8 . Suppose that f (x) = x4 + 4x3 + 6x2 + 3x + 1, then f (x) is a basic primitive polynomial with degree 4 over R. Let ξ = x + (f (x)). Then ξ is a basic primitive element of R = R[x]/(f (x)). Consider a cyclic code C of length 15 and generated by the polynomial g(x) = x10 + 6x9 + x8 + 6x7 + 3x5 + 7x4 + 4x3 + 7x2 + 5x + 1. Then g(x)|(x15 − 1), which implies that C is a free cyclic code and |C | = 645 . Furthermore, g(x) has ξ, ξ 2 , ξ 3 , ξ 4 , ξ 5 , ξ 6 as its part roots in R[x]. Then, by Theorem 7, we have that dH (C ) ≥ 7. Since dH (4g(x)) = 7, it follows that dH (C ) = 7, i.e. C is a (15, 645, 7) cyclic code over R.
4
Module structure of quasi-cyclic codes over R
A linear code C is called quasi-cyclic (QC) code if it is invariant under T ℓ for some positive integer ℓ. If ℓ = 1, then C is a cyclic code of length n over R. The smallest ℓ such that T ℓ (C ) = C is called the index of C . And we call C a QC code of length nℓ with index ℓ over R. Let R be the ℓth Galois extension of R and v = (v00 , . . . , v0,ℓ−1 , . . . , vn−1,0 , . . . , vn−1,ℓ−1 ) ∈ Rnℓ . Define an R-module isomorphism between Rnℓ and Rn by associating with each ℓ-tuple (vi0 , vi1 , . . . , vi,ℓ−1 ), i = 0, 1, . . . , n − 1, and the element vi ∈ R represented as vi = vi0 + vi1 α + · · · + vℓ−1 αℓ−1 , where the set {1, α, . . . , αℓ−1 } forms an R-basis of R. Then every element in Rnℓ is in one-to-one correspondence with an element in Rn . The operator T ℓ for some element (v00 , v01 , . . . , v0,ℓ−1 , . . . , vn−1,0 , vn−1,1 , . . . , vn−1,ℓ−1 ) ∈ Rnℓ corresponds to the element (vn−1 , v0 , . . . , vn−2 ) of Rn . Indicating the block positions with increasing powers of x, the vector v ∈ Rnℓ can be associated with the polynomial v0 + v1 x + · · · + vn−1 xn−1 ∈ R[x]/(xn − 1). An R[x]/(xn − 1)-module isomorphism between Rnℓ and R[x]/(xn − 1) is defined as χ(v) = v0 + v1 x + · · · + vn−1 xn−1 . In this setting, multiplication by x of any element of R[x]/(xn − 1) is equivalent to applying T ℓ to operate the element of Rnℓ . It follows that there is a one-to-one correspondence between R[x]/(xn − 1)submodule of R[x]/(xn − 1) and the QC code of length nℓ with index ℓ over R. Note that a QC code of length nℓ with index ℓ can also be viewed as an R-submodule of R[x]/(xn − 1) because of the equivalence of Rnℓ and R[x]/(xn − 1). Let C be a QC code of length nℓ with index ℓ over R, and assume that it is generated by elements v1 (x), v2 (x), . . . , vr (x) ∈ R[x]/(xn −1) as an R[x]/(xn −1)-submodule of R[x]/(xn − 1). Then C = {a1 (x)v1 (x) + a2 (x)v2 (x) + · · · + ar (x)vr (x) : ai (x) ∈ R[x]/(xn − 1), i = 11
1, 2, . . . , r}. For an R-submodule of R[x]/(xn − 1), C is generated by the following set {v1 (x), xv1 (x), . . . , xn−1 v1 (x), . . . , vr (x), xvr (x), . . . , xn−1 vr (x)}. If C is generated by a single element v(x) as an R[x]/(xn −1)-submodule of R[x]/(xn −1), then C is called the 1-generator QC code. Let the preimage of v(x) in Rnℓ be v under the map χ. Then for a 1-generator QC code C , we have C is generated by the set {v, T ℓ v, . . . , T ℓ(n−1) v}. In fact, let v(x) = v0 + v1 x + · · · + vn−1 xn−1 be a polynomial in R[x]/(xn − 1), where vi = vi0 + vi1 α + · · · + vi,ℓ−1 αℓ−1 , i = 0, 1, . . . , n − 1. Then v(x) becomes an ℓ-tuple of polynomials over R each of degree at most n − 1 with the fixed Rbasis {1, α, . . . , αℓ−1 }. Therefore, v(x) becomes an element of (R[x]/(xn − 1))ℓ . So C is an R[x]/(xn − 1)-submodule of (R[x]/(xn − 1))ℓ . Since R[x]/(xn − 1) is a subring of R[x]/(xn − 1) and C is an R[x]/(xn − 1)-submodule of R[x]/(xn −1), it is in particular a submodule of an R[x]/(xn −1)-submodule of R[x]/(xn −1), i.e. the cyclic code Ce of length n over R. Therefore dH (C ) ≥ dH (Ce). The next result extends Lally’s relevant result [8] to the ring R and its proof is the same, hence is omitted. Theorem 8. Let C be an r-generator QC code of length nℓ with index ℓ over R and generated by the set {v1 (x), v2 (x), . . . , vr (x)}, where vi (x) ∈ R[x]/(xn − 1), i = 1, 2, . . . , r. Then C has a lower bound on the minimum Hamming distance given by dH (C ) ≥ dH (Ce)dH (B), where Ce is the cyclic code of length n over R with generator polynomials v1 (x), v2 (x), . . ., vr (x), and B is a linear code of length ℓ generated by the set {Vij , i = 1, 2, . . . , r, j = 0, 1, . . . , n − 1} ⊆ Rℓ where each Vij is the vector equivalent of the j-th coefficient of vi (x) with respect to an R-basis {1, α, . . . , αℓ−1 }. In fact, Theorem 8 leads to a method to construct QC codes over Zq . Let ℓ = 2. Consider a cyclic code Ce of length n generated by a polynomial v(x) over R. Let C be a linear code of length nℓ spanned by the set {v(x), xv(x), . . . , xn−1 v(x)} over Zq . Then C is a 1-generator QC code of length 2n with index 2. If v(x) = v0 + v1 x + · · · + vn−1 xn−1 ∈ R[x]/(xn − 1), then each vi is a 2-tuple with respect to the fixed Zq -basis {1, u} of R. Now let the set {v0 , v1 , . . . , vn−1 } generate a linear code B of length 2 over Zq . We have the following result directly. Theorem 9. Let C be a QC code of length 2n with index 2 over Zq generated by the set {v(x), xv(x), . . . , xn−1 v(x)}, where v(x) = v0 + v1 x + · · · + vn−1 xn−1 ∈ R[x]/(xn − 1). Then (1) C has a lower bound on the minimum Hamming distance given by dH (C ) ≥ dH (Ce)dH (B), where Ce is a cyclic code of length n over R generated by the polynomial v(x) ∈ R[x]/(xn −1), and B is a linear code of length 2 generated by {v0 , v1 , . . . , vn−1 } where each vi is a 2-tuple with respect to a fixed Zq -basis {1, u} of R. 12
(2) If the cyclic code Ce in (1) is free and the generator polynomial v(x) has δ − 1 consecutive roots in some Galois extension ring of R, and if the set {v0 , v1 , . . . , vn−1 } generates a non-trivial free cyclic code B over Zq of length 2, then dH (C ) ≥ 2δ.
5
1-generator quasi-cyclic codes over R
In Section 4, we have that a 1-generator QC code of length ℓn with index ℓ is an R[x]/(xn −1)submodule of R[x]/(xn − 1) generated by a single element v(x) ∈ R[x]/(xn − 1), where R = GR(R, ℓ), i.e. C = {a(x)v(x) : a(x) ∈ R[x]/(xn − 1)}. As a Zq -submodule of R[x]/(xn − 1), C can be generated by the following set {v(x), xv(x), . . . , xn−1 v(x)}. Let v(x), t(x) ∈ R[x]/(xn − 1). Following the approach given in [3], we use the notation gcdR (v(x), t(x)) for the monic divisor of largest degree of gcd(v(x), t(x)) which lies in R[x]. In fact, gcd(v(x), t(x)) is the largest common monic factor of v(x) and t(x) when they are expressed as ℓ-tuples of polynomials over R. Theorem 10. Let C be a 1-generator QC code of length nℓ with index ℓ over R and generated by v(x) ∈ R[x]/(xn − 1). Let g(x) = gcdR (v(x), xn − 1) and deg(g(x)) = k. Then C is generated by the set {v(x), xv(x), . . . , xn−k−1 v(x)}. Proof. Since g(x) = gcdR (v(x), xn −1), it follows that g(x) is a monic polynomial over R with g(x)|(xn − 1). Therefore xn − 1 = g(x)h(x) for some monic polynomial h(x) over R. Since deg(g(x)) = k, then deg(h(x)) = n−k. Further, for any c(x) ∈ C , there is a polynomial a(x) such that c(x) = a(x)v(x), which implies that c(x) = a(x)v(x) = (q(x)h(x) + r(x))v(x) = r(x)v(x) in R[x]/(xn − 1) with deg(r(x)) ≤ n − k − 1 or r(x) = 0. It means that c(x) can be expressed as an R-combinations of v(x), xv(x), . . . , xn−k−1 v(x). Thus, as an R-submodule of R[x]/(xn − 1), C is generated by the set {v(x), xv(x), . . . , xn−k−1 v(x)}. Review that, by Section 4, a QC code can also be viewed as an R[x]/(xn − 1)-submodule of (R[x]/(xn − 1))ℓ . Let C be a 1-generator QC code generated by the element v(x) = (v0 (x), v1 (x), . . . , vℓ−1 (x)) of (R[x]/(xn − 1))ℓ . For each i = 0, 1, . . . , ℓ − 1, define a surjective R[x]/(xn − 1)-module homomorphism as follows Πi : (R[x]/(xn − 1))ℓ → R[x]/(xn − 1) (v0 (x), v1 (x), . . . , vℓ−1 ) 7→ vi (x). Clearly, if C is a 1-generator QC code of length nℓ with index ℓ generated by the element (v0 (x), v1 (x), . . . , vℓ−1 (x)) of (R[x]/(xn − 1))ℓ then Πi (C ) = (vi (x)) is a cyclic code of length n over R. 13
Lemma 7. Let C = (f (x)) be a free cyclic code of length n with f (x)|(xn − 1) over R. Then C = (m(x)) for some m(x) ∈ R[x] if and only if there exists a polynomial a(x) ∈ R[x] n −1 ) = 1. such that m(x) = a(x)f (x) and gcd(a(x), xf (x) Proof. Let xn −1 = f (x)h(x). Since (f (x)) = (m(x)), then there are polynomial b(x), r(x) ∈ R[x] such that f (x) = m(x)b(x) + r(x)(xn − 1) = a(x)f (x)b(x) + r(x)f (x)h(x). Since f (x) is monic, it follows that 1 = a(x)b(x) + r(x)h(x). Thus, gcd(a(x), h(x)) = 1. On the other hand, if gcd(a(x), h(x)) = 1 then there are polynomials b(x), r(x) ∈ R[x] such that a(x)b(x) + r(x)h(x) = 1. Therefore f (x)a(x)b(x) + f (x)r(x)h(x) = f (x), i.e. m(x)b(x) = f (x) in R[x]/(xn − 1). Thus (f (x)) ⊆ (m(x)). Clearly, (m(x)) ⊆ (f (x)) implying (f (x)) = (m(x)). Theorem 11. Let C be a 1-generator QC code of length nℓ with index ℓ generated by v(x) = (v0 (x), v1 (x), . . . , vℓ−1 (x)) over R with respect to a fixed basis {1, α, . . . , αℓ−1 } of R over R. For each i = 0, 1, . . . , ℓ − 1, if the cyclic code Πi (C ) = (vi (x)) has a monic divisor of xn − 1 as its generator polynomial then C is a free R-submodule of (R[x]/(xn − 1))ℓ . Let g(x) = gcdR (v(x), xn − 1). Then the set {v(x), xv(x), . . . , xn−deg(g(x))−1 v(x)} forms an R-basis of C . Proof. Since gcd(vi (x), xn − 1) = fi (x), it follows that g(x) = gcd(f0 (x), f1 (x), . . . , fℓ−1 (x)) where Πi (C ) = (vi (x)) = (fi (x)) with fi (x)|(xn − 1) for each i = 0, 1, . . . , ℓ − 1. Let P a(x) = n−k−1 ai xi and a(x)v(x) = 0. Then (xn − 1)|a(x)vi (x) for each i = 0, 1, . . . , ℓ − 1. i=0 n n −1 −1 ) = 1. It means that xfi (x) |a(x), Therefore (xn − 1)|a(x)fi (x)ai (x) with gcd(ai (x), xfi (x) xn −1 xn −1 which implies that g(x) |a(x). Since deg( g(x) ) = n − k > deg(a(x)) = n − k − 1, it follows that a(x) = 0. Thus v(x), xv(x), . . . , xn−deg(g(x))−1 v(x) are R-linear independent. Further, v(x), xv(x), . . . , xn−deg(g(x))−1 v(x) generate C by Theorem 10. Therefore the set {v(x), xv(x), . . . , xn−deg(g(x))−1 v(x)} forms an R-basis of C . Theorem 12. Let C be a 1-generator QC code of length nℓ with index ℓ generated by v(x) = (a0 (x)g(x), a1 (x)g(x), . . . , aℓ−1 (x)g(x)) over R, where g(x)|(xn − 1), ai (x) ∈ R[x]/(xn − n −1 ) = 1 for each i = 0, 1, . . . , ℓ − 1. Let the roots of g(x) include 1) and gcd(ai (x), xg(x) b b+1 b+δ−2 ξ ,ξ ,...,ξ , where ξ is a primitive nth root of unity in some Galois extension ring of R. Then dH (C ) ≥ ℓδ. Proof. From Lemma 7, Πi (C ) = (g(x)) for each i = 0, 1, . . . , ℓ − 1. Let c(x) be a codeword of C . Then there exists a polynomial a(x) ∈ R[x] such that c(x) = a(x)(a0 (x)g(x), a1 (x)g(x), . . . , aℓ−1 (x)g(x)).
14
n
−1 . Then h(x)|a(x)ai (x). Let a(x)ai (x)g(x) = 0. Then (xn −1)|a(x)ai (x)g(x). Let h(x) = xg(x) Since gcd(h(x), ai (x)) = 1, it follows that h(x)|a(x). Further, since deg(h(x)) > deg(a(x)), then a(x) = 0 actually. It means that if a(x)ai (x)g(x) = 0 then c(x) = 0. Equivalently, if c(x) 6= 0, then a(x)ai (x)g(x) are all nonzero for all i = 0, 1, . . . , ℓ − 1. Therefore if c(x) is a nonzero codeword of C , then every component of c(x) is a nonzero codeword of the cyclic code (g(x)). From Theorem 7, dH (C ) ≥ ℓδ.
Example 3. Let C = (a0 (x)g(x), a1 (x)g(x) be a 1-generator QC code of length 15 × 2 = 30 with index 2 over R = Z8 + uZ8 , where a0 (x) = x, a1 (x) = x2 and g(x) = x10 + 6x9 + x8 + 6x7 + 3x5 + 7x4 + 4x3 + 7x2 + 5x + 1. From Theorem 12 and Example 2, we have that dH (C ) ≥ 2 × 7 = 14. In fact, dH (C ) = 14. Further, (xg(x), x2 g(x)), (x2 g(x), x3 g(x)), (x3 g(x), x4 g(x)), (x4 g(x), x5 g(x)) and (x5 g(x), x6 g(x)) form a set of R-basis of C . Therefore C is a (30, 645 , 14) QC code over R.
6
Decompositions and duals of quasi-cyclic codes
Since xn − 1 = f1 (x)f2 (x) · · · ft (x), it follows that (R[x]/(xn − 1))ℓ = Rℓ1 ⊕ Rℓ2 ⊕ · · · ⊕ Rℓt and (R[x]/(xn − 1))ℓ ∼ = (R[x]/(f1 (x)))ℓ × (R[x]/(f2 (x)))ℓ × · · · × (R[x]/(ft (x)))ℓ . Then we have that for any QC code C of length nℓ with index ℓ, C can be expressed as C = C1 ⊕ C2 ⊕ · · · ⊕ Ct , where for any l = 1, 2, . . . , ℓ, Cl is a linear code of length ℓ over R[x]/(fl (x)). For any a = (a0 , a1 , . . . , an−1 ) and b = (b0 , b1 , . . . , bn−1 ) of Rn , define the dot product of a and b as a·b=
n−1 X
ai b i .
i=0
The dual of the QC code C of length nℓ with index ℓ over R is defined as C ⊥ = {d ∈ Rnℓ : d · c = 0, for any c ∈ C }. For any a(x) = (a0 (x), a1 (x), . . . , aℓ−1 ) and b(x) = (b0 (x), b1 (x), . . . , bℓ−1 ) of (R[x]/(xn − 1))ℓ , the Hermitian inner product of a(x) and b(x) is defined as a(x) ∗ b(x) =
ℓ−1 X
aj (x)bj (x−1 ).
j=0
Similar to QC codes over finite chain rings [9], we also have the following result on QC codes over R. 15
Theorem 13. Let C be a QC code of length nℓ with index ℓ over R. In other words, C can be viewed as an R[x]/(xn − 1)-submodule of (R[x]/(xn − 1))ℓ . Let C ⊥ is the dual of C . Then C ⊥ is precisely the set {d(x) ∈ (R[x]/(xn − 1))ℓ : d(x) ∗ c(x) = 0, for any c(x) ∈ C }. Further, if C = C1 ⊕ C2 ⊕ · · · ⊕ Ct then C ⊥ = C1⊥ ⊕ C2⊥ ⊕ · · · ⊕ Ct⊥ . In [10], the number of generators of QC codes and their duals over finite fields is given. Following Theorem 6.1 and Corollary 6.3 of [10], we can also get the following result that is similar to Proposition 7 in [3]. Theorem 14. Let C = C1 ⊕ C2 ⊕ · · · ⊕ Ct be a QC code of length nℓ with index ℓ over R, where Cl is a free linear code of length ℓ with rank kl over R[x]/(fl (x)), l = 1, 2, . . . , t. Then C is a K-generator QC code and C ⊥ is an (ℓ−K ′)-generator QC code, where K = maxl (kl ) and K ′ = minl (kl ).
Acknowledgments This research is supported by the National Key Basic Research Program of China (Grant No. 2013CB834204), and the National Natural Science Foundation of China (Grant Nos. 61171082 and 61301137).
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On cyclic codes and quasi-cyclic codes over Zq + uZq Jian Gao∗1 , Fang-Wei Fu1 , Ling Xiao2 , Rama Krishna Bandi3 1. Chern Institute of Mathematics and LPMC, Nankai University Tianjin, 300071, P. R. China 2. School of Science, Shandong University of Technology Zibo, 255091, P. R. China 3. Department of Mathematics,Indian Institute of Technology Roorkee Roorkee, 247667, India E-mail: [email protected]
Abstract Let R = Zq + uZq , where q = ps and u2 = 0. In this paper, some structural properties of cyclic codes and quasi-cyclic (QC) codes over the ring R are considered. A QC code of length ℓn with index ℓ over R is viewed both as in the conventional row circulant form and also as an R[x]/(xn − 1)-submodule of GR(R, ℓ)[x]/(xn − 1), where GR(R, ℓ) is the Galois extension ring of degree ℓ over R. A necessary and sufficient condition for cyclic codes over the ring R to be free is obtained and a BCH-type bound for them is also given. A sufficient condition for 1-generator QC codes to be R-free is given. Some distance bounds for 1-generator QC codes are also discussed. The decomposition and dual of QC codes over R are also briefly discussed. Keywords Cyclic codes; Quasi-cyclic codes; Hensel lift; 1-Generator quasi-cyclic codes
Mathematics Subject Classification (2000) 11T71 · 94B05 · 94B15
1
Introduction
Quasi-cyclic (QC) codes are an important class of linear codes and have some good algebra structures [3]-[10]. Recently, there are some research papers about QC codes over finite chain rings [1, 3, 4, 5, 6, 12]. Aydin et al. studied QC codes over Z4 and obtained some new binary codes using the usual Gray map [1]. Moreover, they characterized cyclic codes corresponding to free modules in terms of their generator polynomials. Bhaintwal et al. discussed QC codes over the prime integer residue ring Zq [3]. They viewed a QC code of length mℓ with index ℓ as an Zq [x]/(xm − 1)-submodule of GR(q, ℓ)[x]/(xm − 1), where GR(q, ℓ) was the ℓth Galois extension ring of Zq . A sufficient condition for 1-generator QC code to be Zq -free was given and some distance bounds for 1-generator QC codes were also discussed. Cui et al. considered 1-generator QC codes over Z4 [4]. Under some conditions,
1
they gave the enumeration of quaternary 1-generator QC codes of length mℓ with index ℓ, and described an algorithm to obtain one and only one generator for each 1-generator QC code. Based on the idea in [4], 1-generator QC codes over another special finite chain ring Fq + uFq were considered [5], and then were generalized to the general finite chain rings [6]. However, it should be noted that all research papers discussed above base on one fact that the block length of the QC code is coprime with the characteristic of finite chain rings. Siap, et al. studied 1-generator QC codes of arbitrary lengths over finite chain ring F2 + uF2 [12]. They gave the generating set and the free condition of the 1-generator QC code. Using Gray map, they also got some optimal binary linear codes over finite field F2 . In these studies, the group rings associated with QC codes are finite chain rings. Recently, Zhu et al. considered linear codes over the finite non-chain ring Fq + vFq [15, 16]. In [16], they studied the cyclic codes over F2 + vF2 . It has shown that cyclic codes over this ring are principally generated. In the subsequent paper [15], they investigated a class of constacyclic codes over Fp + vFp . In that paper, the authors proved that the image of a (1 − 2v)-constacyclic code of length n over Fp + vFp under the Gray map is a cyclic code of length 2n over Fp . Furthermore, they also asserted that (1 − 2v)-constacyclic codes over Fp + vFp are also principally generated. More recently, Yildiz and Karadeniz [14] studied the linear codes over the non-principal ring Z4 + uZ4 , where u2 = 0. They introduced the MacWilliams identities for the complete, symmetrized and Lee weight enumerators. They also gave three methods to construct formally self-dual codes over Z4 + uZ4 . Bandi and Bhaintwal studied some structural properties of cyclic codes of odd length over Z4 + uZ4 , where u2 = 0 [2]. They provided the general form of the generators of a cyclic code over Z4 + uZ4 , and they also determined a necessary condition and a sufficient condition for cyclic codes of odd length over Z4 + uZ4 to be (Z4 + uZ4 )-free. Let R = Zq + uZq , where p is some prime, q = ps and u2 = 0. It is natural to ask if we can also study some structural properties of cyclic codes and QC codes over R. In this paper, we mainly consider this issue. The paper is organized as follows. In Section 2, we introduce some basic results on the ring R. In Section 3, we study the cyclic codes over R. We determine the generator from of cyclic codes. And, we also give a necessary and sufficient condition for cyclic codes over R to be R-free. In Section 4, we discuss module structures of QC codes over R, which studied by the same method of QC codes over finite fields [8] and Galois rings [3]. This point of view for studying QC codes could give a lower bound on the minimum Hamming distance and a construct method of linear codes over Zq . In Section 5, 1-generator QC codes over R are studied. A necessary and sufficient condition for QC cods over R to be R-free is given. Further, we also give a minimum Hamming distance bound on 1-generator QC codes. In Section 6, the decomposition and dual of QC codes are discussed briefly.
2
2
The ring Zq + uZq
Let R = Zq + uZq , where p is some prime, q = ps and u2 = 0. If q = 4, then R = Z4 + uZ4 . The ring R is isomorphic to the quotient ring Zq [u]/(u2 ) and R = {a + bu : a, b ∈ Zq }. Further, The ideals of R are of the following forms: (i) (pi ) for 0 ≤ i ≤ s; (ii) (pk u) for 0 ≤ k ≤ s − 1; (iii) (pj + u) for 1 ≤ j ≤ s − 1; (iv) (pj , u) for 1 ≤ j ≤ s − 1. Therefore, there are 4s − 1 ideals of R. For example, there are 4 × 2 − 1 = 7 ideals of Z4 + uZ4 . R is a local ring with the characteristic q and the maximal ideal (p, u). But it is not a chain ring, since both of the ideals (p) and (u) are not included each other. Further, R is not principal since the ideal (p, u) can not be generated by any single element of this ideal. Define a map −
: R → R/(p, u)
(1)
r = a + bu 7→ a (modp)
The map − is a ring homomorphism and R/(p, u) is denoted by the residue field R. Since for any r = a + bu, a ∈ Zq then R is isomorphic to the finite field Fp . Let R[x] be the polynomial ring over R. The map − can be extended to R[x] to R[x] in the usual way. The image of any element f (x) ∈ R[x] under this map is denoted by f (x). Two polynomials f (x) and g(x) are said to be coprime over R if and only if there are two polynomials a(x) and b(x) in R[x] such that a(x)f (x) + b(x)g(x) = 1.
(2)
A polynomial f (x) ∈ R[x] is said to be basic irreducible if f (x) is irreducible in R[x] and basic primitive if f (x) is primitive in R[x]. In the following, we consider the factorization of xn − 1 over R. We assume that gcd(n, q) = 1 throughout this paper. r
Lemma 1. Let g(x) be a irreducible polynomial over Fp , where g(x)|(xp −1 − 1) for some positive integer r. Then there exists a unique basic irreducible polynomial f (x) such that r f (x) = g(x) and f (x)|(xp −1 − 1) over R. r
Proof. Let xp −1 − 1 = g(x)m(x). Since gcd(n, q) = 1, it follows that g(x) has no multiple roots. Clearly, x ∤ g(x). Then g(x) has the unique Hensel lift f (x) over Zq such that r f (x)|(xp −1 − 1) (see Theorem 13.10 in [13]). Since Zq is a subring of R, it follows that r r the factorization of xp −1 − 1 is still valid over R. It means that f (x)|(xp −1 − 1) over R. Further, g(x) is irreducible over Fp deduces f (x) is basic irreducible over R. 3
We call the polynomial f (x) in Lemma 1 the Hensel lift of g(x) to R[x]. Since gcd(n, q) = 1, it follows that the polynomial xn − 1 can be factored uniquely into pairwise coprime basic irreducible polynomials over R, i.e. xn − 1 = f1 (x)f2 (x) · · · ft (x),
(3)
where, for l = 1, 2, . . . , t, fl (x) is a basic irreducible polynomial over R. m Let T = {0, 1, ξ, . . . , ξ p −2 } be the Teichm¨ uller set of Galois ring GR(q, m), where GR(q, m) is the mth Galois extension ring of Zq and ξ is a basic primitive element of GR(q, m). Then for each a ∈ GR(q, m), it can be written as a = a0 + a1 p + · · · + as−1 ps−1 where a0 , a1 , . . . , as−1 ∈ T . This is called the p-adic representation of the element of GR(q, m) (see the Section 3 of Chapter 14 in [13]). Now we consider the Galois extension of R. Let f (x) be a basic irreducible polynomial of degree m over R. The the mth Galois extension R[x]/(f (x)) is denoted by GR(R, m). Let α be a root of f (x). Then 1, α, . . . , αm−1 form a set of R-free basis and GR(R, m) = {r0 + r1 α + · · · + rm−1 αm−1 : r0 , r1 , . . . , rm−1 ∈ R}.
(4)
The ring GR(R, m) is a local ring with maximal ideal ((p, u) + (f (x))). Its residue field is isomorphic to Fpm . Moreover, GR(R, m) ∼ = GR(q, m)[u]/(u2 ) ∼ = GR(q, m) + uGR(q, m).
(5)
Therefore, for any element r = a + bu ∈ GR(q, m), r=
s−1 X
ai p i + u
Ps−1 i=0
ai p i , b =
b i pi ,
(6)
i=0
i=0
where ai , bi ∈ T and a =
s−1 X
Ps−1 i=0
b i pi .
Ps−1 Ps−1 Lemma 2. For any r = i=0 ai pi + u i=0 bi pi ∈ GR(R, m), r is a unit under multiplication if and only if a0 6= 0. Proof. One can verify that r is a unit under multiplication of R if and only if rk is a unit under multiplication of R for any positive integer k. Particularly, this is valid for k = q. Note that, for any r ∈ R, rq = aq0 ∈ T . Therefore r is a unit if and only if aq0 is a unit if and only if aq0 6= 0 if and only if a0 6= 0. From Lemma 2, we have that the group of units of GR(R, m) denoted by GR(R, m)∗ is given by s−1 s−1 X X bi pi : ai , bi ∈ T , a0 6= 0}. (7) GR(R, m)∗ = { ai p i + u s=0
i=0
Let m
GC = {1, ξ, . . . , ξ p 4
−2
}
and GA = {1 +
s−1 X
j
aj p + u
s−1 X
bi pi : aj , bi ∈ T }.
i=0
j=1
Theorem 1. GR(R, m)∗ = GC × GA , and |GR(R, m)∗ | = (pm − 1) · (p2s−1 )m . m
Proof. Let GC = {1, ξ, . . . , ξ p −2 }. Then GC is a multiplicative cyclic group of order Ps−1 Ps−1 i Ps−1 Ps−1 i pm − 1. Let r = i=0 ai pi + u i=0 bi p be a unit of R, i.e. r = i=0 ai pi + u i=0 bi p ∈ GR(R, m)∗ . Define a group homomorphism as follows Γ : GR(R, m)∗ → GC r=
s−1 X
ai p i + u
s−1 X
bi pi 7→ a0 .
i=0
i=0
Clearly, Γ is a surjective map. Then, by the first homomorphic theorem, we have GR(R, m)∗ /KerΓ ∼ = GC . Clearly, KerΓ = {1 +
s−1 X j=1
j
aj p +
s−1 X
bi pi : a1 , a2 , . . . , as−1 , b0 , b1 , . . . , bs−1 ∈ T }.
i=0
Denote KerΓ by GA . Then GR(R, m)∗ /GA ∼ = GC and |GR(R, m)∗ | = |GC | · |GA | = (pm − 2s−1 m 1) · (p ) The set of all zero divisors of GR(R, m) is {
s−1 X
aj p j + u
s−1 X
bi pi },
(8)
i=0
j=1
which is the maximal ideal of GR(R, m). Lemma 3. Let f (x) and g(x) be polynomials of R[x]. Then f (x) and g(x) are coprime if and only if f (x) and g(x) are coprime over R = Fp . Proof. If f (x) and g(x) are coprime over R, then there are polynomials a(x), b(x) ∈ R[x] such that a(x)f (x) + b(x)g(x) = 1, which implies that a(x)f (x) + b(x)g = 1 with a(x), b(x), f (x), g(x) ∈ Fp [x]. Therefore f (x) and g(x) are coprime over Fp .
5
On the other hand, if f (x) and g(x) are coprime over Fp , then there are polynomials a(x) and b(x) ∈ Fp [x] such that a(x)f (x) + b(x)g = 1, which implies that a(x)f (x) + b(x)g(x) = 1 + pr(x) + ut(x) for some p(x), t(x) ∈ R[x]. Let λ(x) =
s−1 X
(−pr(x))i and τ (x) = 1 − ut(x)λ(x).
i=0
Let κ(x) = λ(x)τ (x). Then κ(x)a(x)f (x) + κ(x)b(x)g(x) = 1, which implies that f (x) and g(x) are coprime over R. Theorem 2. Let GR(R, m) = R[x]/(f (x)) be the mth Galois extension of R, where f (x) is a basic irreducible polynomial with degree m over R. Then the ideals of GR(R, m) are precisely (i) (pi + (f (x))) for 0 ≤ i ≤ s; (ii) (pk u + (f (x))) for 0 ≤ k ≤ s − 1; (iii) (pj + u + (f (x))) for 1 ≤ j ≤ s − 1; (iv) ((pj , u) + (f (x))) for 1 ≤ j ≤ s − 1. Proof. Let I be an ideal of R. If I is zero, then I = (ps + (f (x))) of R[x]/(f (x)). In the following, we determine the nonzero ideals of R[x]/(f (x)). Let g(x) ∈ I. Since f (x) is basic irreducible over R, it follows that f (x) is irreducible over Fp . Therefore gcd(f (x), g(x)) = 1 or f (x). If gcd(f (x), g(x)) = 1, then gcd(f (x), g(x)) = 1 which implies that there are polynomials a(x), b(x) ∈ R[x] such that a(x)f (x) + bg(x) = 1. It means that g(x) is a unit of R[x]/(f (x)), i.e. I = R[x]/(f (x)). If gcd(f (x), g(x)) = f (x), then there are polynomials a(x), b(x), c(x) ∈ R[x] such that g(x) = a(x)f (x) + pb(x) + uc(x). Therefore g(x) ∈ ((p, u) + (f (x))) of R[x]/(f (x)). Since the ideals contained in g(x) ∈ ((p, u) + (f (x))) are as the form in this theorem. The result follows.
3
Cyclic codes over R
Let Rn be a free R-module of rank n, i.e. Rn = {(c0 , c1 , . . . , cn−1 ) : c0 , c1 , . . . , cn−1 ∈ R}. Let C be a nonempty set of Rn . C is called a linear code of length n if and only if C is an R-submodule of Rn . Let T be the cyclic shift operator. If for any c = (c0 , c1 , . . . , cn−1 ) ∈ C
6
the T (c) = (cn−1 , c0 , . . . , cn−2 ) is also in C , we say C is a cyclic code of length n over R. Define an R-module isomorphism as follows Φ : Rn → R[x]/(xn − 1) (c0 , c1 , . . . , cn−1 ) 7→ c0 + c1 x + · · · + cn−1 xn−1 . One can verify that C is a cyclic code of length n over R if and only if Φ(C ) is an ideal of the quotient ring R[x]/(xn − 1). Sometimes, we identity the cyclic code of length n over R with the ideal of R[x]/(xn − 1). Review that xn − 1 = f1 (x)f2 (x) · · · ft (x), where, for each l = 1, 2, . . . , t, fl (x) is a n −1 by fbl (x). Since fl (x) and fbl (x) are basic irreducible polynomial over R. Denote xfl (x) coprime to each other, it follows that there are polynomials al (x), bl (x) in R[x] such that al (x)fl (x) + bl (x)fbl (x) = 1. Let el (x) = bl (x)fbl (x) + (xn − 1) and Ri = el (x)R[x]/(xn − 1). Then we have R[x]/(xn − 1) = R1 ⊕ R1 ⊕ · · · ⊕ Rt . (9) For any l = 1, 2, . . . , t, the map Ψ : R[x]/(fl (x)) → Rl k(x) + (fl (x)) 7→ (k(x) + (xn − 1))el (x)
(10)
is an isomorphism of rings. Therefore, R[x]/(xn − 1) ∼ = R[x]/(f1 (x)) × R[x]/(f2 (x)) × · · · × R[x]/(ft (x)).
(11)
Lemma 4. Let xn − 1 = f1 (x)f2 (x) · · · ft (x) where, for each l = 1, 2, . . . , t, fl (x) is a basic irreducible polynomial over R. Then under the map Ψ, the ideals of R[x]/(fl (x)) are mapped into (i) (pi fbl (x) + (xn − 1) for 0 ≤ i ≤ s; (ii) (pk ufbl (x) + (xn − 1)) for 0 ≤ k ≤ s − 1; (iii) ((pj + u)fbl (x) + (xn − 1)) for 1 ≤ j ≤ s − 1; (iv) ((pj , u)fbl (x) + (xn − 1)) for 1 ≤ j ≤ s − 1 of Rl . Proof. Under the ring isomorphism Ψ, we have 1 + (fl (x)) 7→ (1 + (xn − 1))el . Since el = bl (x)fbl (x) + (xn − 1), it follows that 1 + (fl (x)) 7→ bl (x)fbl (x) + (xn − 1). Clearly, bl (x)fbl (x) + (xn − 1) ∈ (fbl (x) + (xn − 1)). 7
Multiplying both sides of al (x)fl (x) + bl (x)fbl (x) = 1 by fbl (x), we obtain bl (x)fbl (x)fbl (x) + al (x)(xn − 1) = fbl (x). Then bl (x)fbl (x)fbl (x) + (xn − 1) = fbl (x) + (xn − 1),
which implies that fbl (x) + (xn − 1) ∈ (bl (x)fbl (x) + (xn − 1)). Therefore, (bl (x)fbl (x) + (xn − 1)) = (fbl (x) + (xn − 1)) and the image of (1 + (fi (x))) under the ring isomorphism Ψ is (fbl (x) + (xn − 1)). The remainder cases can also be verified in the same way. By Theorem 2, Eq. (11) and Lemma 4, we have the following result directly. Theorem 3. Let xn − 1 = f1 (x)f2 (x) · · · ft (x) where, for each l = 1, 2, . . . , t, fl (x) is a basic irreducible polynomial over R. Then there are (4s − 1)t cyclic codes of length n over R. Further, any cyclic code is the sum of the ideals of Rl . Example 1. Consider a cyclic code of length 3 over Z4 + uZ4 . Since x3 − 1 = (x − 1)(x2 + x + 1), it follows that there are (4 × 2 − 1)2 = 72 = 49 cyclic codes of length 3 over Z4 + uZ4 . Let f1 = x − 1 and f2 = x2 + x + 1. In the following Table 1, we list all of cyclic codes of length 3 over Z4 + uZ4 . Table 1: All cyclic codes of length 3 over Z4 + uZ4 0 (f1 ) (2f1 ) (uf1 ) (2uf1 ) ((2 + u)f1 ) ((2, u)f2 ) ((2 + u)f2 ) (2 + u) ((2, u)f1 , uf2 )
(f2 ) (1) (2f1 , f2 ) (uf1 , f2 ) (2uf1 , f2 ) ((2 + u)f1 , f2 ) (f1 , (2, u)f2 ) (f1 , (2 + u)f2 ) ((2 + u)f1 , (2, u)f2 ) ((2, u)f1 , 2uf2 )
(2f2 ) (f1 , 2f2 ) (2) (uf1 , 2f2 ) (2uf1 , 2f2 ) ((2 + u)f1 , 2f2 ) (2f1 , (2, u)f2 ) (2uf1 , (2 + u)f2 ) ((2, u)f1 ) ((2, u)f1 , (2 + u)f2 )
(uf2 ) (f1 , uf2 ) (2f1 , uf2 ) (u) (2uf1 , uf2 ) ((2 + u)f1 , uf2 ) (uf1 , (2, u)f2 ) (2f1 , (2 + u)f2 ) ((2, u)f1 , f2 ) (2, u)
(2uf2 ) (f1 , 2uf2 ) (2f1 , 2uf2 ) (uf1 , 2uf2 ) (2u) ((2 + u)f1 , 2uf2 ) (2uf1 , (2, u)f2 ) (uf1 , (2 + u)f2 ) ((2, u)f1 , 2f2 )
In fact, the cyclic code C also has the following general form of generators. Theorem 4. Let C be a cyclic code of length n over R. Then C = (f0 (x) + uf1 (x), ug1 (x)) with f0 (x), f1 (x), g1 (x) ∈ Zq [x] and g1 (x)|f0 (x). Proof. Define a surjective homomorphism from R to Zq as ψ(a + bu) = a for any a + bu ∈ R. Extend ψ to the polynomial ring R[x]/(xn − 1) as ψ(a0 + a1 + · · · + an−1 xn−1 ) = ψ(a0 ) +
8
ψ(a1 )x + · · · + ψ(an−1 )xn−1 for any polynomial a0 + a1 x + · · · + an−1 ∈ R[x]/(xn − 1). Let C be a cyclic code of length n over R, and restrict ψ to C . Define a set J = {f (x) ∈ Zq [x]/(xn − 1) : uf (x) ∈ Kerψ}. Clearly, J is an ideal of Zq [x]/(xn −1), which implies that there is a polynomial g1 (x) ∈ Zq [x] such that J = (g1 (x)). It means that Kerψ = (ug1 (x)). Further, the image of C under the map ψ is also an ideal of Zq [x]/(xn − 1). Then there is a polynomial f0 (x) ∈ Zq [x] such that ψ(C ) = (f0 (x)). Hence C = (f0 (x) + uf1 (x), ug1 (x)). Clearly, u(f0 (x) + uf1 (x)) = uf0 (x) ∈ Kerψ implying g0 (x)|f0 (x). Lemma 5. Let C = (f0 (x) + uf1 (x), ug1 (x)) be a cyclic code of length n over R. If g1 (x) is a monic polynomial over R, then we can assume that deg(f1 (x)) < deg(g1 (x)). Proof. If deg(g1 (x)) ≤ deg(f1 (x)), then there are polynomials m(x), r(x) ∈ R[x] such that f1 (x) = g1 (x)m(x) + r(x) with deg(r(x)) < deg(g1 (x)) or r(x) = 0. Then C = (f0 (x) + ug1 (x)m(x) + ur(x), ug1 (x)). Let C1 = (f0 (x) + ur(x), ug1 (x)). Clearly, C ⊆ C1 . Further, f0 (x) + ur(x) = f0 (x) + ug1 (x)m(x) + ur(x) + u(q − 1)g1 (x)m(x), it follows that C1 ⊆ C . Thus C = C1 . Lemma 6. Let C = (f0 (x) + uf1 (x), ug1 (x)) be a cyclic code of length n over R. If f0 (x) = g1 (x), then C = (f0 (x) + uf1 (x)). Further, if g1 (x) is monic over R, then (f0 (x) + uf1 (x))|(xn − 1). Proof. Clearly, (f0 (x) + uf1 (x)) ⊆ C . Further, since u(f0 (x) + uf1 (x)) = uf0 (x) = ug1 (x), it follows that C ⊆ (f0 (x) + uf1 (x)). Thus C = (f0 (x) + uf1 (x)). Since f0 (x) = g1 (x) and g1 is monic, then f0 (x) + uf1 (x) is also monic over R by Lemma 5. Therefore there are polynomials a(x), b(x) ∈ R[x] such that xn − 1 = a(x)(f0 (x) + uf1 (x)) + b(x) with b(x) = 0 or deg(b(x)) < deg(f0 (x)). Since b(x) ∈ C , it follows that b(x) = 0. Thus (f0 (x) + uf1 (x))|(xn − 1) over R. Theorem 5. Let C = (f0 (x) + uf1 (x), ug1 (x)) be a cyclic code of length n over R and f0 (x), g1 (x) be monic over Zq . Let deg(f0 (x)) = k0 and deg(g1 (x)) = k1 . Then the set β = {(f0 (x) + uf1 (x)), . . . , xn−k0 −1 (f0 (x) + uf1 (x)), ug1 (x), . . . , xk0 −k1 −1 ug1 (x)} forms the minimum generating set of C , and |C | = q 2n−k0 −k1 . 9
Proof. Let γ = {(f0 (x) + uf1 (x)), . . . , xn−k0 −1 (f0 (x) + uf1 (x)), ug1 (x), . . . , xn−k1 −1 ug1 (x)}. Then γ spans the cyclic code C . Further, it is sufficient to show that β spans γ, which follows that β also spans C . Now we only need to show that β is linearly independent. For simplicity, we denote f (x) as f0 (x) + uf1 (x). Since f0 is monic, then the constant coefficient of f (x) is a unit of R by the fact that f0 (x) is the generator polynomial of some cyclic code of length n over Zq . Let a0 f (x) + a1 xf (x) + · · · + an−k0 −1 xn−k0 −1 f (x) = 0.
(12)
Let F0 be the constatant coefficient of f (x). Then a0 F0 = 0, which implies that a0 = 0. Therefore the Eq. (12) becomes a1 f (x)+a2 xf (x)+· · ·+an−k0 −1 xn−k0 −2 f (x) = 0. Similarly, a1 = a2 = · · · = an−k0 −1 = 0. Thus (f0 (x)+uf1 (x)), x(f0 (x)+uf1 (x)), . . . , xn−k0 −1 (f0 (x)+ uf1 (x)) are R-linear independent. One can also prove that ug1 (x), uxg1 (x), . . . , uxn−k1 −1 g1 (x) are Zq -linear independent. Thus |C | = q 2n−k0 −k1 . We now consider the cyclic code C as a principal ideal of R[x]/(xn − 1). Theorem 6. Let C be a principal generated cyclic code of length n over R. Then C is free if and only if there is a monic polynomial g(x) such that C = (g(x)) and g(x)|(xn − 1). Moreover, the set {g(x), xg(x), . . . , xn−deg(g(x))−1 g(x)} forms the minimum generating set of C and |C | = q 2(n−deg(g(x))) . Proof. Suppose that C = (g(x)) is R-free. Then the set {g(x), xg(x), . . . , xn−deg(g(x))−1 g(x)} form the R-basis of C . Since xn−degg(x) ∈ C , it follows that xn−deg(g(x)) can be written as a linear combination of the elements g(x), xg(x), . . . , xn−deg(g(x))−1 g(x), i.e. xn−deg(g(x)) g(x)+ Pn−deg(g(x))−1 P ai xi g(x) = 0. Let a(x) = n−deg(g(x))−1 ai xi +xn−deg(g(x)) . Then a(x)g(x) = i=0 i=0 0 in R[x]/(xn − 1), which implies that (xn − 1)|g(x)a(x). Since g(x)a(x) is monic and deg(g(x)a(x)) = n, it follows that xn − 1 = g(x)a(x) implying g(x)|(xn − 1). On the other hand, if C = (g(x)) with g(x)|(xn −1). Then g(x), xg(x), . . . , xn−deg(g(x))−1 g(x) span C . In the following, we prove that g(x), xg(x), . . . , xn−deg(g(x))−1 g(x) are R-linear independent. Let g(x) = g0 + g1 x + · · · + xdeg(g(x)) and a0 g(x) + a1 xg(x) + · · · + an−deg(g(x))−1 xn−deg(g(x))−1 g(x) = 0.
(13)
Since g0 is a unit of R and a0 g0 = 0, it follows that a0 = 0. Then the Eq. (13) becomes a1 g(x) + a2 xg(x) + · · · + an−deg(g(x))−1 xn−deg(g(x))−2 g(x) = 0. Similarly, we have a1 = a2 = · · · = an−deg(g(x))−1 = 0. Thus g(x), xg(x), . . . , xn−deg(g(x))−1 g(x) are R-linear independent, i.e. C is free over R.
10
Theorem 7. (BCH-type Bound) Let C = (g(x)) be a free cyclic code of length n over R. Suppose that g(x) has roots ξ b , ξ b+1 , . . . , ξ b+δ−2 , where ξ is a basic primitive nth root of unity in some Galois extension ring of R. Then the minimum Hamming distance of C is dH (C ) ≥ δ. Proof. The proof process is similar to that of Proposition 2 in [3]. Example 2. Let R = Z8 + uZ8 . Suppose that f (x) = x4 + 4x3 + 6x2 + 3x + 1, then f (x) is a basic primitive polynomial with degree 4 over R. Let ξ = x + (f (x)). Then ξ is a basic primitive element of R = R[x]/(f (x)). Consider a cyclic code C of length 15 and generated by the polynomial g(x) = x10 + 6x9 + x8 + 6x7 + 3x5 + 7x4 + 4x3 + 7x2 + 5x + 1. Then g(x)|(x15 − 1), which implies that C is a free cyclic code and |C | = 645 . Furthermore, g(x) has ξ, ξ 2 , ξ 3 , ξ 4 , ξ 5 , ξ 6 as its part roots in R[x]. Then, by Theorem 7, we have that dH (C ) ≥ 7. Since dH (4g(x)) = 7, it follows that dH (C ) = 7, i.e. C is a (15, 645, 7) cyclic code over R.
4
Module structure of quasi-cyclic codes over R
A linear code C is called quasi-cyclic (QC) code if it is invariant under T ℓ for some positive integer ℓ. If ℓ = 1, then C is a cyclic code of length n over R. The smallest ℓ such that T ℓ (C ) = C is called the index of C . And we call C a QC code of length nℓ with index ℓ over R. Let R be the ℓth Galois extension of R and v = (v00 , . . . , v0,ℓ−1 , . . . , vn−1,0 , . . . , vn−1,ℓ−1 ) ∈ Rnℓ . Define an R-module isomorphism between Rnℓ and Rn by associating with each ℓ-tuple (vi0 , vi1 , . . . , vi,ℓ−1 ), i = 0, 1, . . . , n − 1, and the element vi ∈ R represented as vi = vi0 + vi1 α + · · · + vℓ−1 αℓ−1 , where the set {1, α, . . . , αℓ−1 } forms an R-basis of R. Then every element in Rnℓ is in one-to-one correspondence with an element in Rn . The operator T ℓ for some element (v00 , v01 , . . . , v0,ℓ−1 , . . . , vn−1,0 , vn−1,1 , . . . , vn−1,ℓ−1 ) ∈ Rnℓ corresponds to the element (vn−1 , v0 , . . . , vn−2 ) of Rn . Indicating the block positions with increasing powers of x, the vector v ∈ Rnℓ can be associated with the polynomial v0 + v1 x + · · · + vn−1 xn−1 ∈ R[x]/(xn − 1). An R[x]/(xn − 1)-module isomorphism between Rnℓ and R[x]/(xn − 1) is defined as χ(v) = v0 + v1 x + · · · + vn−1 xn−1 . In this setting, multiplication by x of any element of R[x]/(xn − 1) is equivalent to applying T ℓ to operate the element of Rnℓ . It follows that there is a one-to-one correspondence between R[x]/(xn − 1)submodule of R[x]/(xn − 1) and the QC code of length nℓ with index ℓ over R. Note that a QC code of length nℓ with index ℓ can also be viewed as an R-submodule of R[x]/(xn − 1) because of the equivalence of Rnℓ and R[x]/(xn − 1). Let C be a QC code of length nℓ with index ℓ over R, and assume that it is generated by elements v1 (x), v2 (x), . . . , vr (x) ∈ R[x]/(xn −1) as an R[x]/(xn −1)-submodule of R[x]/(xn − 1). Then C = {a1 (x)v1 (x) + a2 (x)v2 (x) + · · · + ar (x)vr (x) : ai (x) ∈ R[x]/(xn − 1), i = 11
1, 2, . . . , r}. For an R-submodule of R[x]/(xn − 1), C is generated by the following set {v1 (x), xv1 (x), . . . , xn−1 v1 (x), . . . , vr (x), xvr (x), . . . , xn−1 vr (x)}. If C is generated by a single element v(x) as an R[x]/(xn −1)-submodule of R[x]/(xn −1), then C is called the 1-generator QC code. Let the preimage of v(x) in Rnℓ be v under the map χ. Then for a 1-generator QC code C , we have C is generated by the set {v, T ℓ v, . . . , T ℓ(n−1) v}. In fact, let v(x) = v0 + v1 x + · · · + vn−1 xn−1 be a polynomial in R[x]/(xn − 1), where vi = vi0 + vi1 α + · · · + vi,ℓ−1 αℓ−1 , i = 0, 1, . . . , n − 1. Then v(x) becomes an ℓ-tuple of polynomials over R each of degree at most n − 1 with the fixed Rbasis {1, α, . . . , αℓ−1 }. Therefore, v(x) becomes an element of (R[x]/(xn − 1))ℓ . So C is an R[x]/(xn − 1)-submodule of (R[x]/(xn − 1))ℓ . Since R[x]/(xn − 1) is a subring of R[x]/(xn − 1) and C is an R[x]/(xn − 1)-submodule of R[x]/(xn −1), it is in particular a submodule of an R[x]/(xn −1)-submodule of R[x]/(xn −1), i.e. the cyclic code Ce of length n over R. Therefore dH (C ) ≥ dH (Ce). The next result extends Lally’s relevant result [8] to the ring R and its proof is the same, hence is omitted. Theorem 8. Let C be an r-generator QC code of length nℓ with index ℓ over R and generated by the set {v1 (x), v2 (x), . . . , vr (x)}, where vi (x) ∈ R[x]/(xn − 1), i = 1, 2, . . . , r. Then C has a lower bound on the minimum Hamming distance given by dH (C ) ≥ dH (Ce)dH (B), where Ce is the cyclic code of length n over R with generator polynomials v1 (x), v2 (x), . . ., vr (x), and B is a linear code of length ℓ generated by the set {Vij , i = 1, 2, . . . , r, j = 0, 1, . . . , n − 1} ⊆ Rℓ where each Vij is the vector equivalent of the j-th coefficient of vi (x) with respect to an R-basis {1, α, . . . , αℓ−1 }. In fact, Theorem 8 leads to a method to construct QC codes over Zq . Let ℓ = 2. Consider a cyclic code Ce of length n generated by a polynomial v(x) over R. Let C be a linear code of length nℓ spanned by the set {v(x), xv(x), . . . , xn−1 v(x)} over Zq . Then C is a 1-generator QC code of length 2n with index 2. If v(x) = v0 + v1 x + · · · + vn−1 xn−1 ∈ R[x]/(xn − 1), then each vi is a 2-tuple with respect to the fixed Zq -basis {1, u} of R. Now let the set {v0 , v1 , . . . , vn−1 } generate a linear code B of length 2 over Zq . We have the following result directly. Theorem 9. Let C be a QC code of length 2n with index 2 over Zq generated by the set {v(x), xv(x), . . . , xn−1 v(x)}, where v(x) = v0 + v1 x + · · · + vn−1 xn−1 ∈ R[x]/(xn − 1). Then (1) C has a lower bound on the minimum Hamming distance given by dH (C ) ≥ dH (Ce)dH (B), where Ce is a cyclic code of length n over R generated by the polynomial v(x) ∈ R[x]/(xn −1), and B is a linear code of length 2 generated by {v0 , v1 , . . . , vn−1 } where each vi is a 2-tuple with respect to a fixed Zq -basis {1, u} of R. 12
(2) If the cyclic code Ce in (1) is free and the generator polynomial v(x) has δ − 1 consecutive roots in some Galois extension ring of R, and if the set {v0 , v1 , . . . , vn−1 } generates a non-trivial free cyclic code B over Zq of length 2, then dH (C ) ≥ 2δ.
5
1-generator quasi-cyclic codes over R
In Section 4, we have that a 1-generator QC code of length ℓn with index ℓ is an R[x]/(xn −1)submodule of R[x]/(xn − 1) generated by a single element v(x) ∈ R[x]/(xn − 1), where R = GR(R, ℓ), i.e. C = {a(x)v(x) : a(x) ∈ R[x]/(xn − 1)}. As a Zq -submodule of R[x]/(xn − 1), C can be generated by the following set {v(x), xv(x), . . . , xn−1 v(x)}. Let v(x), t(x) ∈ R[x]/(xn − 1). Following the approach given in [3], we use the notation gcdR (v(x), t(x)) for the monic divisor of largest degree of gcd(v(x), t(x)) which lies in R[x]. In fact, gcd(v(x), t(x)) is the largest common monic factor of v(x) and t(x) when they are expressed as ℓ-tuples of polynomials over R. Theorem 10. Let C be a 1-generator QC code of length nℓ with index ℓ over R and generated by v(x) ∈ R[x]/(xn − 1). Let g(x) = gcdR (v(x), xn − 1) and deg(g(x)) = k. Then C is generated by the set {v(x), xv(x), . . . , xn−k−1 v(x)}. Proof. Since g(x) = gcdR (v(x), xn −1), it follows that g(x) is a monic polynomial over R with g(x)|(xn − 1). Therefore xn − 1 = g(x)h(x) for some monic polynomial h(x) over R. Since deg(g(x)) = k, then deg(h(x)) = n−k. Further, for any c(x) ∈ C , there is a polynomial a(x) such that c(x) = a(x)v(x), which implies that c(x) = a(x)v(x) = (q(x)h(x) + r(x))v(x) = r(x)v(x) in R[x]/(xn − 1) with deg(r(x)) ≤ n − k − 1 or r(x) = 0. It means that c(x) can be expressed as an R-combinations of v(x), xv(x), . . . , xn−k−1 v(x). Thus, as an R-submodule of R[x]/(xn − 1), C is generated by the set {v(x), xv(x), . . . , xn−k−1 v(x)}. Review that, by Section 4, a QC code can also be viewed as an R[x]/(xn − 1)-submodule of (R[x]/(xn − 1))ℓ . Let C be a 1-generator QC code generated by the element v(x) = (v0 (x), v1 (x), . . . , vℓ−1 (x)) of (R[x]/(xn − 1))ℓ . For each i = 0, 1, . . . , ℓ − 1, define a surjective R[x]/(xn − 1)-module homomorphism as follows Πi : (R[x]/(xn − 1))ℓ → R[x]/(xn − 1) (v0 (x), v1 (x), . . . , vℓ−1 ) 7→ vi (x). Clearly, if C is a 1-generator QC code of length nℓ with index ℓ generated by the element (v0 (x), v1 (x), . . . , vℓ−1 (x)) of (R[x]/(xn − 1))ℓ then Πi (C ) = (vi (x)) is a cyclic code of length n over R. 13
Lemma 7. Let C = (f (x)) be a free cyclic code of length n with f (x)|(xn − 1) over R. Then C = (m(x)) for some m(x) ∈ R[x] if and only if there exists a polynomial a(x) ∈ R[x] n −1 ) = 1. such that m(x) = a(x)f (x) and gcd(a(x), xf (x) Proof. Let xn −1 = f (x)h(x). Since (f (x)) = (m(x)), then there are polynomial b(x), r(x) ∈ R[x] such that f (x) = m(x)b(x) + r(x)(xn − 1) = a(x)f (x)b(x) + r(x)f (x)h(x). Since f (x) is monic, it follows that 1 = a(x)b(x) + r(x)h(x). Thus, gcd(a(x), h(x)) = 1. On the other hand, if gcd(a(x), h(x)) = 1 then there are polynomials b(x), r(x) ∈ R[x] such that a(x)b(x) + r(x)h(x) = 1. Therefore f (x)a(x)b(x) + f (x)r(x)h(x) = f (x), i.e. m(x)b(x) = f (x) in R[x]/(xn − 1). Thus (f (x)) ⊆ (m(x)). Clearly, (m(x)) ⊆ (f (x)) implying (f (x)) = (m(x)). Theorem 11. Let C be a 1-generator QC code of length nℓ with index ℓ generated by v(x) = (v0 (x), v1 (x), . . . , vℓ−1 (x)) over R with respect to a fixed basis {1, α, . . . , αℓ−1 } of R over R. For each i = 0, 1, . . . , ℓ − 1, if the cyclic code Πi (C ) = (vi (x)) has a monic divisor of xn − 1 as its generator polynomial then C is a free R-submodule of (R[x]/(xn − 1))ℓ . Let g(x) = gcdR (v(x), xn − 1). Then the set {v(x), xv(x), . . . , xn−deg(g(x))−1 v(x)} forms an R-basis of C . Proof. Since gcd(vi (x), xn − 1) = fi (x), it follows that g(x) = gcd(f0 (x), f1 (x), . . . , fℓ−1 (x)) where Πi (C ) = (vi (x)) = (fi (x)) with fi (x)|(xn − 1) for each i = 0, 1, . . . , ℓ − 1. Let P a(x) = n−k−1 ai xi and a(x)v(x) = 0. Then (xn − 1)|a(x)vi (x) for each i = 0, 1, . . . , ℓ − 1. i=0 n n −1 −1 ) = 1. It means that xfi (x) |a(x), Therefore (xn − 1)|a(x)fi (x)ai (x) with gcd(ai (x), xfi (x) xn −1 xn −1 which implies that g(x) |a(x). Since deg( g(x) ) = n − k > deg(a(x)) = n − k − 1, it follows that a(x) = 0. Thus v(x), xv(x), . . . , xn−deg(g(x))−1 v(x) are R-linear independent. Further, v(x), xv(x), . . . , xn−deg(g(x))−1 v(x) generate C by Theorem 10. Therefore the set {v(x), xv(x), . . . , xn−deg(g(x))−1 v(x)} forms an R-basis of C . Theorem 12. Let C be a 1-generator QC code of length nℓ with index ℓ generated by v(x) = (a0 (x)g(x), a1 (x)g(x), . . . , aℓ−1 (x)g(x)) over R, where g(x)|(xn − 1), ai (x) ∈ R[x]/(xn − n −1 ) = 1 for each i = 0, 1, . . . , ℓ − 1. Let the roots of g(x) include 1) and gcd(ai (x), xg(x) b b+1 b+δ−2 ξ ,ξ ,...,ξ , where ξ is a primitive nth root of unity in some Galois extension ring of R. Then dH (C ) ≥ ℓδ. Proof. From Lemma 7, Πi (C ) = (g(x)) for each i = 0, 1, . . . , ℓ − 1. Let c(x) be a codeword of C . Then there exists a polynomial a(x) ∈ R[x] such that c(x) = a(x)(a0 (x)g(x), a1 (x)g(x), . . . , aℓ−1 (x)g(x)).
14
n
−1 . Then h(x)|a(x)ai (x). Let a(x)ai (x)g(x) = 0. Then (xn −1)|a(x)ai (x)g(x). Let h(x) = xg(x) Since gcd(h(x), ai (x)) = 1, it follows that h(x)|a(x). Further, since deg(h(x)) > deg(a(x)), then a(x) = 0 actually. It means that if a(x)ai (x)g(x) = 0 then c(x) = 0. Equivalently, if c(x) 6= 0, then a(x)ai (x)g(x) are all nonzero for all i = 0, 1, . . . , ℓ − 1. Therefore if c(x) is a nonzero codeword of C , then every component of c(x) is a nonzero codeword of the cyclic code (g(x)). From Theorem 7, dH (C ) ≥ ℓδ.
Example 3. Let C = (a0 (x)g(x), a1 (x)g(x) be a 1-generator QC code of length 15 × 2 = 30 with index 2 over R = Z8 + uZ8 , where a0 (x) = x, a1 (x) = x2 and g(x) = x10 + 6x9 + x8 + 6x7 + 3x5 + 7x4 + 4x3 + 7x2 + 5x + 1. From Theorem 12 and Example 2, we have that dH (C ) ≥ 2 × 7 = 14. In fact, dH (C ) = 14. Further, (xg(x), x2 g(x)), (x2 g(x), x3 g(x)), (x3 g(x), x4 g(x)), (x4 g(x), x5 g(x)) and (x5 g(x), x6 g(x)) form a set of R-basis of C . Therefore C is a (30, 645 , 14) QC code over R.
6
Decompositions and duals of quasi-cyclic codes
Since xn − 1 = f1 (x)f2 (x) · · · ft (x), it follows that (R[x]/(xn − 1))ℓ = Rℓ1 ⊕ Rℓ2 ⊕ · · · ⊕ Rℓt and (R[x]/(xn − 1))ℓ ∼ = (R[x]/(f1 (x)))ℓ × (R[x]/(f2 (x)))ℓ × · · · × (R[x]/(ft (x)))ℓ . Then we have that for any QC code C of length nℓ with index ℓ, C can be expressed as C = C1 ⊕ C2 ⊕ · · · ⊕ Ct , where for any l = 1, 2, . . . , ℓ, Cl is a linear code of length ℓ over R[x]/(fl (x)). For any a = (a0 , a1 , . . . , an−1 ) and b = (b0 , b1 , . . . , bn−1 ) of Rn , define the dot product of a and b as a·b=
n−1 X
ai b i .
i=0
The dual of the QC code C of length nℓ with index ℓ over R is defined as C ⊥ = {d ∈ Rnℓ : d · c = 0, for any c ∈ C }. For any a(x) = (a0 (x), a1 (x), . . . , aℓ−1 ) and b(x) = (b0 (x), b1 (x), . . . , bℓ−1 ) of (R[x]/(xn − 1))ℓ , the Hermitian inner product of a(x) and b(x) is defined as a(x) ∗ b(x) =
ℓ−1 X
aj (x)bj (x−1 ).
j=0
Similar to QC codes over finite chain rings [9], we also have the following result on QC codes over R. 15
Theorem 13. Let C be a QC code of length nℓ with index ℓ over R. In other words, C can be viewed as an R[x]/(xn − 1)-submodule of (R[x]/(xn − 1))ℓ . Let C ⊥ is the dual of C . Then C ⊥ is precisely the set {d(x) ∈ (R[x]/(xn − 1))ℓ : d(x) ∗ c(x) = 0, for any c(x) ∈ C }. Further, if C = C1 ⊕ C2 ⊕ · · · ⊕ Ct then C ⊥ = C1⊥ ⊕ C2⊥ ⊕ · · · ⊕ Ct⊥ . In [10], the number of generators of QC codes and their duals over finite fields is given. Following Theorem 6.1 and Corollary 6.3 of [10], we can also get the following result that is similar to Proposition 7 in [3]. Theorem 14. Let C = C1 ⊕ C2 ⊕ · · · ⊕ Ct be a QC code of length nℓ with index ℓ over R, where Cl is a free linear code of length ℓ with rank kl over R[x]/(fl (x)), l = 1, 2, . . . , t. Then C is a K-generator QC code and C ⊥ is an (ℓ−K ′)-generator QC code, where K = maxl (kl ) and K ′ = minl (kl ).
Acknowledgments This research is supported by the National Key Basic Research Program of China (Grant No. 2013CB834204), and the National Natural Science Foundation of China (Grant Nos. 61171082 and 61301137).
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