EVALUATION OF THE WEIGHT DISTRIBUTION OF A CLASS OF ...
EVALUATION OF THE WEIGHT DISTRIBUTION OF A CLASS OF LINEAR CODES BASED ON SEMI-PRIMITIVE GAUSS SUMS
arXiv:1511.02093v2 [cs.IT] 18 Nov 2015
ZILING HENG AND QIN YUE Abstract. Linear codes with a few weights have been widely investigated in recent years. However, most of these known linear codes were defined over a prime field. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of q-ary linear codes, where q is a power of a prime. The lower bound of the minimum Hamming distance of the codes is obtained. And in some special cases, we evaluate the weight distributions of the linear codes based on semi-primitive Gauss sums and obtain some one-weight, two-weight linear codes. It is quite interesting that we find a few optimal codes achieving some certain bounds on linear codes. The linear codes in this paper can be used in secret sharing schemes, authentication codes and data storage systems.
1. Introduction Let Fq denote the finite field with q elements, where q is a power of a prime p. An [n, l, d] linear code C over Fq is a l-dimensional subspace of Fnq with minimum Hamming distance d. There are some bounds on linear codes. Let nq (l, d) be the minimum length n for which an [n, l, d] linear code over Fq exists. The well-known Griesmer bound is given by l−1 X d nq (l, d) ≥ ⌈ i ⌉. q i=0 The Singleton bound is given by
nq (l, d) ≥ l + d − 1. An [n, l, d] code is called optimal if no [n, l, d + 1] code exists, and is called almost optimal if the [n, l, d + 1] code is optimal. Let Ai denote the number of codewords with Hamming weight i in a code C with length n. The weight enumerator of C is defined by 1 + A1 z + · · · + An z n . The sequence (A1 , A2 , · · · , An ) is called the weight distribution of C. The code C is called to be t-weight if the number of nonzero Aj , 1 ≤ j ≤ n, in the sequence 2010 Mathematics Subject Classification. 11T71, 11T55. Key words and phrases. linear codes, weight distribution, secret sharing schemes. The paper is supported by NNSF of China (No. 11171150) Fundamental Research Funds for the Central Universities (No. NZ2015102); Funding of Jiangsu Innovation Program for Graduate Eduction (the Fundamental Research Funds for the Central Universities; No. KYZZ15 0086). 1
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Z. HENG AND Q. YUE
(A1 , A2 , · · · , An ) equals to t. Weight distribution is an interesting topic and was investigated by [1, 4, 10, 11, 17, 18, 20, 21, 27, 28]. It could be used to estimate the error-correcting capability and the error probability of error detection of a code. Let D = {d1 , d2 , . . . , dn } ⊆ Fr , where r is a power of q. Let Trr/q be the trace function from Fr onto Fq . A linear code of length n over Fq is defined by CD = {(Trr/q (xd1 ), Trr/q (xd2 ), . . . , Trr/q (xdn )) : x ∈ Fr }. The set D is called the defining set of CD . Although different orderings of the elements of D result in different codes CD , these codes are permutation equivalent and have the same length, dimension and weight distribution. Hence, the orderings of the elements of D will not affect the results in this correspondence. If the set D is well chosen, the code CD may have good parameters. This construction is generic in the sense that many known codes [5, 6, 7, 8, 12, 13, 17, 22, 24, 26, 29, 30, 31, 32, 34] could be produced by selecting the defining set. However, most of these known codes focused on linear codes over a prime field. Let Trqk /q , Trqf /q denote the trace functions from Fqk to Fq and Fqf to Fq , respectively. Let f, k be positive integers such that f |k. In this paper, a class of q-ary linear codes CD is defined by CD = {(Trqk /q (xd1 ), Trqk /q (xd2 ), . . . , Trqk /q (xdn )) : x ∈ Fqk } F∗qk
(1.1)
q k −1 q f −1
) + a = 0}, where a ∈ Fq . Let Nqk /qf with the defining set D = {x ∈ : Trqf /q (x be the norm function from Fqk to Fqf . In fact, the defining set D is constructed from q k −1
the composite function Trqf /q ◦Nqk /qf due to Trqf /q (x qf −1 ) = Trqf /q (Nqk /qf (x)). We investigate this class of linear codes in the following cases: (1) a = 0, f > 1; (2) a ∈ F∗q , gcd( fk , q − 1) = 1. We use Gauss sums to represent their Hamming weights and obtain the lower bounds of their minimum distances. For f = 2 in Case (1) and f = 1, 2 in Case (2), the weight distributions of the linear codes are explicitly determined. Some codes with one or two weights are obtained. In particular, we obtain some codes which are optimal or almost optimal with respect to some certain bounds on linear codes. Two-weights codes are closely related to strongly regular graphs, partial geometries and projective sets [14, 15]. Linear codes with a few weights have applications in secret sharing schemes [25, 33] and authentication codes [9]. For convenience, we introduce the following notations in this paper:
LINEAR CODES
primitive element of Fqk ,
α β=α χ χ1 χ2 ψ ψ1 ψ2
3
q k −1 q f −1
primitive element of Fqf , canonical additive character of Fq , canonical additive character of Fqf , canonical additive character of Fqk , multiplicative character of Fq , multiplicative character of Fqf , multiplicative character of Fqk . 2. Gauss sums
Let Fq be a finite field with q elements, where q is a power of a prime p. The canonical additive character of Fq is defined as follows: Trq/p (x)
χ : Fq −→ C∗ , χ(x) = ζp
,
where ζp denotes the p-th primitive root of unity and Trq/p is the trace function from Fq to Fp . The orthogonal property of additive characters [23] is given by: ( X q, if a = 0, χ(ax) = 0 otherwise. x∈F q
Let ψ : F∗q −→ C∗ be a multiplicative character of F∗q . The trivial multiplicative character ψ0 is defined by ψ0 (x) = 1 for all x ∈ F∗q . It is known [23] that all the b∗ , which is isomorphic to F∗ . multiplicative characters form a multiplication group F q q The orthogonal property of a multiplicative character ψ is given by (see [23]): ( X q − 1, if ψ = ψ0 , ψ(x) = 0 otherwise. x∈F∗ q
The Gauss sum over Fq is defined by G(ψ, χ) =
X
ψ(x)χ(x).
x∈F∗q
¯ χ) = ψ(−1)G(ψ, χ). If ψ 6= ψ0 , we It is easy to see that G(ψ0 , χ) = −1 and G(ψ, √ have |G(ψ, χ)| = q. Gauss sum is an important tool in this paper to compute exponential sums. In general, the explicit determination of Gauss sums is a difficult problem. In some cases, Gauss sums are explicitly determined in [3, 10, 23]. The semi-primitive case Gauss sums, which will be used in this correspondence, are known as follows. Lemma 2.1. [3] Let λ be a multiplicative character of order N of F∗r and ρ the canonical additive character of Fr . Assume that N 6= 2 and their exists a least
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Z. HENG AND Q. YUE
positive integer j such that pj ≡ −1 (mod N). Let r = p2jγ for some integer γ. Then the Gauss sums of order N over Fr are given by ( √ if p = 2, (−1)γ−1 r, G(λ, ρ) = γ(pj +1) √ γ−1+ N (−1) r, if p ≥ 3. Furthermore, for 1 ≤ s ≤ N − 1, the Gauss sums G(ψ s ) are given by ( √ j (−1)s r, if N is even, p, γ and p N+1 are odd, s G(λ , ρ) = √ (−1)γ−1 r, otherwise. 3. The case a = 0 Let f be a positive integer such that f |k and k > f > 1. In this section, we investigate CD defined in (1.1) with the defining set q k −1
D = {d1 , d2 , · · · , dn } = {x ∈ F∗qk : Trqf /q (x qf −1 ) = 0}. To determine the length n of CD , we need a lemma below. Lemma 3.1. For the canonical additive character χ1 of Fqf , we have X X
x∈Fqk y∈Fq
χ1 (yx
q k −1 q f −1
) = q + qk − 1 −
(q k − 1)(q − 1) . qf − 1
Proof. For 0 ≤ j ≤ q k −2, let j = (q f −1)s+t with 0 ≤ s ≤ Then X X
x∈Fqk y∈Fq
q k −1
χ1 (yx qf −1 ) = q + q k − 1 +
X X
q k −1 −1 q f −1
and 0 ≤ t ≤ q f −2.
q k −1
χ1 (yx qf −1 )
x∈F∗k y∈F∗q q
k
k
= q+q −1+
−2 X qX
= q + qk − 1 +
−1
f
q −2 −1 X qfX X
y∈F∗q
q k −1
χ1 (yα qf −1
((q f −1)s+t)
)
t=0
s=0
qk − 1 X X χ1 (yz) q f − 1 y∈F∗ z∈F∗ q
= q + qk − 1 −
j
y∈F∗q j=0
q k −1
= q + qk − 1 +
q k −1
χ1 (yα qf −1 )
qf
k
(q − 1)(q − 1) . qf − 1
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5
q k −1
Let n0 = |{x ∈ Fqk : Trqf /q (x qf −1 ) = 0}|. Note that χ1 = χ ◦ Trqf /q . Then q k −1 1 X X q χ(y Trqf /q (x f −1 )) = q x∈F y∈F
n0
q
qk
q k −1 1 X X χ1 (yx qf −1 ) q x∈F y∈F
=
q
qk
= 1+ by Lemma 3.1. Hence,
For each b ∈ F∗qk , let
(q k − 1)(q f − q) q(q f − 1)
(q k − 1)(q f − q) n = n0 − 1 = . q(q f − 1)
(3.1)
q k −1
Nb = |{x ∈ Fqk : Trqf /q (x qf −1 ) = 0 and Trqk /q (bx) = 0}|. By the basic facts of additive characters, for each b ∈ F∗qk we have Nb
X q k −1 1 X X q = 2 χ(Trqf /q (yx f −1 ))( ( χ(Trqk /q (bzx)) q x∈F y∈F z∈F q
qk
=
q
X q k −1 1 X X q f −1 ))( ( χ (yx χ2 (bzx)) 1 q 2 x∈F y∈F z∈F q
qk
= q k−2 +
q
1 X X 1 X X q f −1 )) + χ (yx χ2 (bzx)) ( ( 1 q 2 x∈F y∈F∗ q 2 x∈F z∈F∗ q k −1
qk
qk
q
q
q k −1 1 X XX χ1 (yx qf −1 )χ2 (bzx). + 2 q x∈F y∈F∗ z∈F∗ qk
q
q
By Lemma 3.1, we have X X X X q k −1 q k −1 χ1 (yx qf −1 )) ( χ1 (yx qf −1 )) = −q k + ( x∈Fqk y∈Fq
x∈Fqk y∈F∗q
=
(q − 1)(q f − q k ) . qf − 1
By the orthogonal relation of additive characters, we have X X X X χ2 (bzx) = 0. χ2 (bzx)) = ( x∈Fqk z∈F∗q
Let Ω(b) :=
P P
q k −1
z∈F∗q x∈Fqk
χ1 (yx qf −1 )χ2 (bx) and ∆(b) :=
x∈Fqk y∈F∗q
Nb = q k−2 +
P
Ω(bz). Then we have
z∈F∗q
(q − 1)(q f − q k ) 1 + 2 ∆(b). 2 f q (q − 1) q
(3.2)
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Z. HENG AND Q. YUE
In the following, we compute the exponential sum ∆(b), b ∈ F∗qk . Lemma 3.2. [23] Let r be a power of a prime p. Let ρ be a nontrivial additive character of Fr , m ∈ N, and λ a multiplicative character of Fr of order s = gcd(m, r− 1). Then s−1 X X m ¯ j (a0 )G(λj , ρ) ρ(a0 c + a1 ) = ρ(a1 ) λ j=1
c∈Fr
for any a0 , a1 ∈ Fr and a0 6= 0.
Lemma 3.3. [23] (Davenport-Hasse Theorem) Let r be a power of a prime p. Let ρ be an additive and λ a multiplicative character of Fr , not both of them trivial. Suppose ρ and λ are lifted to characters ρ′ and λ′ , respectively, of the extension field E of Fr with [E : Fr ] = t. Then G(λ′ , ρ′ ) = (−1)t−1 G(λ, ρ)t . Lemma 3.4. Let χ1 , χ2 be the canonical additive characters of Fqf and Fqk , respectively. Let f be a positive integer such that f |k. Then for b ∈ F∗qk , X
χ2 (bxq
f −1
k
) = (−1) f −1
x∈F∗k
X
q k −1
k G(ψ1 , χ1 ) f ψ¯1 (b qf −1 ).
b∗ ψ1 ∈F f
q
q
Proof. By Lemma 3.2, we have X X f f χ2 (bxq −1 ) χ2 (bxq −1 ) = −1 + x∈Fqk
x∈F∗k q
q f −2
= −1 + q f −2
X
=
X
ψ¯2j (b)G(ψ2j , χ2 )
j=1
ψ¯2j (b)G(ψ2j , χ2 ),
j=0
where ψ2 is a multiplicative character of order q f − 1 of Fqk . Let Nqk /qf be the norm mapping from Fqk to Fqf . Then ψ2 = ψ1 ◦ Nqk /qf can be seen as the lift of ψ1 from b∗f to F b ∗k . Note that Nqk /qf is an epimorphism. Then ord(ψ1 ) = ord(ψ2 ) = q f − 1. F q q Therefore, by Lemma 3.3, X X f G(ψ1 ◦ Nqk /qf , χ2 )ψ¯1 (Nqk /qf (b)) χ2 (bxq −1 ) = x∈F∗k q
b∗ ψ1 ∈F f q
k
= (−1) f −1
X
q k −1
k G(ψ1 , χ1 ) f ψ¯1 (b qf −1 ).
b∗ ψ1 ∈F f q
LINEAR CODES
7
Lemma 3.5. Let f be a positive integer such that f > 1 and f |k, then f
2
k −1 f
f
q (q − 1) (−1) q (q − 1) + ∆(b) = f f q −1 q −1
2
q f −1 −1 q−1
X
where ϕ is a multiplicative character of order
j
j
ϕ (−1)G(ϕ , χ1 )
k −1 f
j
ϕ¯ (b
q k −1 q f −1
),
j=1
q f −1 q−1
of Fqf . q k −1
Proof. Let F∗qk = hαi and F∗qf = hβi. Then β = α qf −1 . And there is a coset decomposition as follows: q f −2
F∗qk
[
=
j=0
αj hαq
f −1
i.
Hence, Ω(b) = q − 1 +
X X
q k −1
χ1 (yx qf −1 )χ2 (bx)
x∈F∗k y∈F∗q q
f
= q−1+
−2 X qX
χ1 (yβ j )
y∈F∗q j=0
X
ω∈hαq
f −1
χ2 (bωαj ) i
q f −2
= q−1+
X 1 XX f j χ2 (bxq −1 αj ) χ (yβ ) 1 f q − 1 y∈F∗ j=0 x∈F∗ q
qk
By Lemma 3.4, k X q k −1 k (−1) f −1 X X Ω(b) = (q − 1) + f G(ψ1 , χ1 ) f ψ¯1 (b qf −1 x) χ1 (yx) q − 1 x∈F∗ y∈F∗ ∗ qf
b ψ1 ∈F
q
qf
k X q k −1 k (−1) f −1 X X = (q − 1) + f G(ψ1 , χ1 ) f ψ¯1 (b qf −1 y −1) ψ¯1 (yx)χ1 (yx) q −1 ∗ y∈F∗ x∈F∗
b ψ1 ∈F
qf
q
qf
k X q k −1 k (−1) f −1 X G(ψ1 , χ1 ) f G(ψ¯1 , χ1 )ψ¯1 (b qf −1 ) ψ1 (y). = (q − 1) + f q −1 ∗ ∗ y∈F
b ψ1 ∈F
qf
q
This implies that X Ω(bz) ∆(b) = z∈F∗q
k X X q k −1 k k (−1) f −1 X ¯ ¯ f q = (q − 1) + f ψ¯1 (z f ). ψ1 (y) G(ψ1 , χ1 ) G(ψ1 , χ1 )ψ1 (b f −1 ) q −1 ∗ z∈F∗ y∈F∗
2
b ψ1 ∈F
qf
q
q
8
Z. HENG AND Q. YUE q f −1
Since the norm function Nqf /q : F∗qf → F∗q , x 7→ y = x q−1 , is an epimorphism, X
q f −1 q−1 X q−1 ). ψ (x 1 q f − 1 x∈F∗
ψ1 (y) =
y∈F∗q
Similarly, we have
qf
X
−1 k k ·f q − 1 X ¯ qq−1 ψ¯1 (z f ) = f ). ψ1 (x1 q − 1 ∗ ∗ z∈F x ∈F f
1
q
qf
Then we have k q k −1 k (−1) f −1 (q − 1)2 X ¯ ¯ f G(ψ q f −1 ) ∆(b) = (q − 1) + G(ψ , χ ) , χ ) ψ (b 1 1 1 1 1 (q f − 1)3 ∗
2
b ψ1 ∈F
X
ψ1 (x
q f −1 q−1
)
X
q f −1 k · q−1 f
ψ¯1 (x1
qf
).
x1 ∈F∗f q
x∈F∗f q
Note that X
ψ1 (x
q f −1 q−1
)=
(
−1 q f − 1, if ord(ψ1 )| qq−1 , 0, otherwise,
)=
(
−1 k q f − 1, if ord(ψ1 )| qq−1 · f, 0, otherwise.
x∈F∗f q
and X
q f −1 k · q−1 f
ψ¯1 (x1
x1 ∈F∗f q
f
f
Hence, we have 2
∆(b) = (q − 1) + f
2
(−1)
k −1 f
(q − 1) f q −1
2
q f −1 −1 q−1
k −1 f
(−1) q (q − 1) (q − 1) + = qf − 1 qf − 1 f
=
k −1 f
q (q − 1) (q − 1) (−1) + qf − 1 qf − 1 f
=
2
2
k −1 f
f
X
k f
j
j
j
G(ϕ , χ1 ) G(ϕ¯ , χ1 )ϕ¯ (b
q k −1 q f −1
)
j=0
2
q f −1 −1 q−1
X
G(ϕ , χ1 ) G(ϕ¯ , χ1 )ϕ¯ (b
X
G(ϕ , χ1 ) f ϕj (−1)G(ϕj , χ1 )ϕ¯j (b qf −1 )
j
k f
j
k
j
j
q k −1 q f −1
)
j=1
2
q f −1 −1 q−1
q (q − 1) q (q − 1) (−1) + f f q −1 q −1
q k −1
j=1
2
q f −1 −1 q−1
X
j
k
q k −1
ϕj (−1)G(ϕ , χ1 ) f −1 ϕ¯j (b qf −1 ),
j=1
where ϕ is a multiplicative character of order
q f −1 q−1
of Fqf .
In general, determining the value distribution of ∆(b) is a very difficult problem. In the following, we determine the value distribution of ∆(b) if f = 2.
LINEAR CODES
9
Lemma 3.6. For an odd integer q, let ζq+1 be the primitive q + 1-th root of complex , we have unity. Then for any integer 1 ≤ s ≤ q and s 6= q+1 2 qs s 3s 5s ζq+1 + ζq+1 + ζq+1 + · · · + ζq+1 = 0,
and (q−1)s
2s 4s 6s ζq+1 + ζq+1 + ζq+1 + · · · + ζq+1
= −1.
qs s 3s 5s Proof. It is clear that {ζq+1 , ζq+1 , ζq+1 , · · · , ζq+1 } is a geometric progression. Hence qs s 3s 5s ζq+1 + ζq+1 + ζq+1 + · · · + ζq+1 2s· q+1
s ζq+1 (1 − ζq+1 2 ) = = 0. 2s 1 − ζq+1
Similarly, (q−1)s
2s 4s 6s ζq+1 + ζq+1 + ζq+1 + · · · + ζq+1 (q−1)s
2s 2s ζq+1 − ζq+1 ζq+1 = −1. = 2s 1 − ζq+1
Lemma 3.7. If f = 2 and f |k, then the value distribution of ∆(b), b ∈ F∗qk , is given as follows: (1) If p = 2, then k k (q−1)(q2 +(−1) 2 −1 q 2 +2 ) , qk −1 times, q+1 q+1 ∆(b) = k k q(q k −1) (q−1)(q2 +(−1) 2 q 2 +1 ) , times. q+1
(2) If p > 2, then q2 (q−1) q+1 k −1 2q+1 (1 + 2 q 2 + k q (q−1) ∆(b) = (1 − q+1 q 2 −1 + q+1 2 k q2 (q−1) (1 − (−q) 2 −1 ), q+1
q+1
k q−1 (−q) 2 −1 ), 2 k q−1 (−q) 2 −1 ), 2
q k −1 times, q+1 q k −1 times, q+1 (q k −1)(q−1) times. q+1
f
−1 Proof. Let f = 2, then qq−1 = q + 1. For the multiplicative character ϕ of order e N = q + 1 = p + 1, G(ϕ, χ1 ) in Lemma 3.5 is just a semi-primitive case Gauss sum over Fq2 by Lemma 2.2. Note that ϕ(−1) = 1 if f = 2. And by the proof of Lemma 3.5, we have k q q k −1 k (−1) 2 −1 (q − 1)2 X j j j q2 −1 2 G(ϕ G(ϕ , χ ) ¯ , χ ) ϕ ¯ (b ) ∆(b) = (q − 1) + 1 1 q2 − 1 j=0
2
(q+1,q 2 )
with ord(ϕ) = q + 1. Let Cs = β s hβ q+1 i, s = 0, 1, · · · , q, be the cyclotomic ¯ = classes of order q + 1 over Fq2 . Without loss of generality, we assume that ϕ(β)
10
Z. HENG AND Q. YUE q k −1
q k −1
(q+1,q 2 )
sj , j = 0, 1, · · · , q, when b q−1 ∈ Cs ζq+1 . It is clear that ϕ¯j (b q−1 ) = ζq+1 q k −1 q−1
0, 1, · · · , q. Choose bs ∈ F∗qk , such that bs ts =
q X
k 2
j
j
(q+1,q 2 )
∈ Cs j
q k −1 q 2 −1
G(ϕ , χ1 ) G(ϕ¯ , χ1 )ϕ¯ (bs
j=0
Hence, ∆(bs ) = (q − 1)2 +
,s =
, s = 0, 1, · · · , q. Denote
), s = 0, 1, · · · , q.
k
(−1) 2 −1 (q−1)2 ts , q 2 −1
q k −1
s = 0, 1, · · · , q. For a fixed 0 ≤ s ≤ q, the
value of ∆(bs ) occurs q+1 times when b runs through F∗qk . (1) Assume that p = 2. Then q is even. Let ϕ¯0 (β 0 ) ϕ¯1 (β 0 ) ϕ¯2 (β 0 ) . . . ϕ¯q−1 (β 0 ) ϕ¯q (β 0 ) ϕ¯0 (β) ϕ¯1 (β) ϕ¯2 (β) . . . ϕ¯q−1 (β) ϕ¯q (β) ϕ¯0 (β 2 ) ϕ¯1 (β 2 ) ϕ¯2 (β 2 ) . . . ϕ¯q−1 (β 2 ) ϕ¯q (β 2 ) T := .. .. .. .. .. .. . . . . . . 0 q−1 1 q−1 2 q−1 q−1 q−1 q ϕ¯ (β ) ϕ¯ (β ) ϕ¯ (β ) . . . ϕ¯ (β ) ϕ¯ (β q−1 ) ϕ¯0 (β q ) ϕ¯1 (β q ) ϕ¯2 (β q ) . . . ϕ¯q−1 (β q ) ϕ¯q (β q ) 1 1 1 ... 1 1 q−1 q 2 1 ζq+1 ζq+1 . . . ζq+1 ζq+1 2(q−1) 2q 4 1 ζ2 ζ . . . ζ ζ q+1 q+1 q+1 q+1 = .. .. .. .. .. .. . . . . . . 1 ζ q−1 ζ 2(q−1) . . . ζ (q−1)2 ζ q(q−1) q+1 q+1 q+1 q+1 (q−1)q q 2q q2 1 ζq+1 ζq+1 . . . ζq+1 ζq+1
(q+1)×(q+1)
(q+1)×(q+1)
which is called the character matrix of Fq2 . And by Lemma 2.5, G(ϕs , χ1 ) = q, 1 ≤ s ≤ q. Hence, k k G(ϕ0 , χ1 ) 2 G(ϕ¯0 , χ1 ) (−1) 2 +1 t0 k 2 k2 −1 G(ϕ, χ1 ) 2 G(ϕ, ¯ χ1 ) q ·q t1 .. .. = T = ... . T . . k 2 k −1 G(ϕq−1 , χ1 ) 2 G(ϕ¯q−1 , χ1 ) q · q 2 tq−1 k k tq q 2 · q 2 −1 G(ϕq , χ1 ) 2 G(ϕ¯q , χ1 ) Note that for 1 ≤ s ≤ q,
qs s 2s ζq+1 + ζq+1 + · · · + ζq+1 = −1.
Hence, we have (
k
k
t0 = (−1) 2 +1 + q 2 +2 , k k ts = (−1) 2 +1 − q 2 +1 , 1 ≤ s ≤ q.
Then the value distribution of ∆(b), b ∈ F∗qk , follows.
LINEAR CODES
(2) Let p > 2. Then q is odd. Let ϕ¯0 (β 0 ) ϕ¯1 (β 0 ) ϕ¯2 (β 0 ) . . . ϕ¯q−1 (β 0 ) ϕ¯q (β 0 ) ϕ¯0 (β) ϕ¯1 (β) ϕ¯2 (β) . . . ϕ¯q−1 (β) ϕ¯q (β) ϕ¯1 (β 2 ) ϕ¯2 (β 2 ) . . . ϕ¯q−1 (β 2 ) ϕ¯q (β 2 ) ϕ¯0 (β 2 ) .. .. .. .. .. .. . . . . . . ′ T := 0 q+1 q+1 q+1 q+1 q+1 ϕ¯ (β 2 ) ϕ¯1 (β 2 ) ϕ¯2 (β 2 ) . . . ϕ¯q−1 (β 2 ) ϕ¯q (β 2 ) .. .. .. .. .. .. . . . . . . 0 q−1 ϕ¯ (β ) ϕ¯1 (β q−1 ) ϕ¯2 (β q−1 ) . . . ϕ¯q−1 (β q−1 ) ϕ¯q (β q−1 ) ϕ¯0 (β q ) ϕ¯1 (β q ) ϕ¯2 (β q ) . . . ϕ¯q−1 (β q ) ϕ¯q (β q ) 1 1 1 ... 1 1 q−1 q 2 1 ζq+1 ζq+1 . . . ζq+1 ζq+1 2(q−1) 2q 2 4 1 ζq+1 ζ . . . ζ ζ q+1 q+1 q+1 .. .. .. .. .. .. . . . . . . = 1 −1 1 . . . 1 −1 .. .. .. .. .. .. . . . . . . 2 2(q−1) (q−1) q(q−1) q−1 ζ . . . ζ 1 ζ ζ q+1 q+1 q+1 q+1 (q−1)q q 2q q2 1 ζq+1 ζq+1 . . . ζq+1 ζq+1 (q+1)×(q+1)
11
(q+1)×(q+1)
which is called the character matrix of Fq2 . And by Lemma 2.5, G(ϕs , χ1 ) = (−1)s q, 1 ≤ s ≤ q. Hence, k k (−1) 2 +1 G(ϕ0 , χ1 ) 2 G(ϕ¯0 , χ1 ) k k 2 G(ϕ, χ1 ) 2 G(ϕ, ¯ χ1 ) q · (−q) 2 −1 t k k 0 q 2 · q 2 −1 G(ϕ2 , χ1 ) 2 G(ϕ¯2 , χ1 ) t1 k q 2 · (−q) k2 −1 G(ϕ3 , χ1 ) 2 G(ϕ¯3 , χ1 ) .. = T′ T′ = . . . . .. .. q 2 · (−q) k2 −1 tq−1 G(ϕq−2 , χ ) k2 G(ϕ¯q−2 , χ ) 1 1 tq k k q−1 2 q−1 −1 2 2 q ·q G(ϕ , χ1 ) G(ϕ¯ , χ1 ) k k q 2 · (−q) 2 −1 G(ϕq , χ1 ) 2 G(ϕ¯q , χ1 ) By Lemma 3.6, we have k k k k (−1) 2 −1 (q+1)q 2 +1 (q−1)q 2 +1 +1 2 + + , t = (−1) 0 2 2 k k +1 +1 ts = (−1) 2 − q 2 , k k k t q+1 = (−1) k2 +1 + (−1) 2 (q+1)q 2 +1 + (q−1)q 2 +1 , 2 2 2
where 1 ≤ s ≤ q and s 6= follows.
q+1 . 2
Then the value distribution of ∆(b), b ∈ F∗qk ,
Theorem 3.8. Let f |k and k > f > 1. Let CD be the linear code defined by (1.1) with a = 0. The for a codeword cb = (Trqk /q (bd1 ), · · · , Trqk /q (bd1 )) ∈ CD , b ∈ F∗qk , the
12
Z. HENG AND Q. YUE
Hamming weight of it equals to k−2
w(cb ) =
k −1 f
f
f
(q − 1)q (q − q) (−1) q (q − 1) − f 2 f q −1 q (q − 1)
where ϕ is a multiplicative character of order
2
q f −1 −1 q−1
X
q k −1
k
j
ϕj (−1)G(ϕ , χ1 ) f −1 ϕ¯j (b qf −1 ),
j=1
q f −1 q−1
over Fqf . And CD is a
(q k − 1)(q f − q) (q − 1)(q f − q)(q k−2 − q [ , k, d ≥ q(q f − 1) qf − 1
k+f −4 2
)
] k−1
k
k −1 2
−1 , k, (q−1)(q q+1−q ) ] linear code. In particular, if f = 2 and k ≡ 0 (mod 4), CD is a [ qq+1 two-weight code with the weight distribution given in Table I; if f = 2 and k ≡ 2 k 2
k−1
k
−1 (mod 4), CD is a [ qq+1 , k, (q−1)(qq+1 −q ) ] two-weight code with the weight distribution given in Table II.
Table I. Weight distribution of CD for f = 2 and k ≡ 0 (mod 4) weight Frequency 0 1 k
(q−1)(q k−1 +q 2 ) q+1 k (q−1)(q k−1 −q 2 −1 ) q+1
q k −1 q+1 q(q k −1) q+1
Table II. Weight distribution of CD for f = 2 and k ≡ 2 weight Frequency 0 1 k
(q−1)(q k−1 −q 2 ) q+1 k (q−1)(q k−1 +q 2 −1 ) q+1
(mod 4)
q k −1 q+1 q(q k −1) q+1
Proof. For a codeword cb = (Trqk /p (bd1 ), · · · , Trqk /p (bd1 )), b ∈ F∗qk , the Hamming weight of it equals to n0 − Nb . Then by Equations (3.1) and (3.2), Lemma 3.5, we have k−2
w(cb ) =
k −1 f
f
f
q (q − 1) (q − 1)q (q − q) (−1) − qf − 1 q 2 (q f − 1)
where ϕ is a multiplicative character of order k −1 f
| ≤ Then we have
f
(−1) q (q − 1) 2 f q (q − 1)
2
2
q f −1 −1 q−1
q f −1 q−1
X j=1
over Fqf . Note that
q f −1 −1 q−1
X
k
j
k
j
q k −1
ϕj (−1)G(ϕ , χ1 ) f −1 ϕ¯j (b qf −1 )|
j=1
p k −1 q f (q − 1)2 q f − 1 ( − 1)( qf ) f q 2 (q f − 1) q − 1 (q − 1)(q f − q)(q k−2 − q w(cb ) ≥ qf − 1
q k −1
ϕj (−1)G(ϕ , χ1 ) f −1 ϕ¯j (b qf −1 ),
k+f −4 2
)
>0
LINEAR CODES
13
due to k > f > 1. Then the dimension of CD is k. For f = 2, the weight distributions can be obtained by Equations (3.1) and (3.2), Lemma 3.7. Example 3.9. Let f = 2, k = 4 and q = 4, CD in Theorem 3.8 is a [51, 4, 36] linear code, which has the same parameters as the best known linear codes according to [16], with weight enumerator 1 + 204z 36 + 51z 48 . This coincides with the result given in Theorem 3.8. Example 3.10. Let f = 2, k = 4 and and q = 9, CD in Theorem 3.8 is a [656, 4, 576] linear code with weight enumerator 1 + 5904z 576 + 656z 648 . This coincides with the result given in Theorem 3.8. It is observed that the weights of CD in Tables I-II have a common divisor q − 1. This indicates that the code CD can be punctured into a shorter code CDe as follows. q k −1
q k −1
k
q k −1
Note that Trqf /q ((ax) qf −1 ) = a f Trqf /q (x q−1 ) = 0 for all a ∈ Fq if Trqf /q (x qf −1 ) = 0. Hence, the defining set of CD defined by (3.1) can be expressed as e = {ade : a ∈ F∗ , de ∈ D}, e D = F∗q D q
(3.3)
e Then by Theorem where dei /dej 6∈ F∗q for every pair of distinct elements dei , dej in D. 3.8, we have the following result. Corollary 3.11. Let f |k and k > f = 2. Let CDe be the linear code with defining set e defined in (3.3). Then C e is a [ qk2 −1 , k] linear code with the weight distributions D D q −1 given in Tables III, IV for k ≡ 0 (mod 4) and k ≡ 2 (mod 4), respectively. Table III. Weight distribution of CDe for f = 2 and k ≡ 0 (mod 4) weight Frequency 0 1 k
q k−1 +q 2 q+1 k q k−1 −q 2 −1 q+1
q k −1 q+1 q(q k −1) q+1
Table IV. Weight distribution of CDe for f = 2 and k ≡ 2 (mod 4) weight Frequency 0 1 k
q k−1 −q 2 q+1 k q k−1 +q 2 −1 q+1
q k −1 q+1 q(q k −1) q+1
Remark 3.12. Let f = 2, k = 4, then CDe in Corollary 3.11 is an optimal [q 2 + 1, 4, q 2 − q] linear code achieving the Griesmer bound.
14
Z. HENG AND Q. YUE
4. The case a ∈ F∗q In this section, we assume that f is a positive integer such that f |k and gcd( fk , q − 1) = 1. Let other notations be the same as that of Section 3. Now we investigate the linear code defined in (1.1) with the defining set q k −1
D = {d1 , d2, · · · , dn } = {x ∈ Fqk : Trqf /q (x qf −1 ) + a = 0}, where a ∈ F∗q . The length of CD equals to n = |D|. We need a lemma given below to obtain the value of n. Lemma 4.1. For the canonical additive character χ of Fq , we have X X
q k −1
χ(y(Trqf /q (x qf −1 ) + a)) =
x∈Fqk y∈Fq
q f (q k − 1) . qf − 1
Proof. The proof is similar to that of Lemma 3.1. We omit it here.
By Lemma 4.1, we have n=
q k −1 1 X X q f −1 (q k − 1) χ(y(Trqf /q (x qf −1 ) + a)) = . q x∈F y∈F qf − 1
(4.1)
q
qk
For each b ∈ F∗qk , let Nb = |{x ∈ Fqk : Trqf /q (x
q k −1 q f −1
) + a = 0 and Trqk /q (bx) = 0}|.
By the basic facts of additive characters and Lemma 4.1, for any b ∈ F∗qk we have Nb =
X q k −1 1 X X q f −1 ) + a)))( χ(y(Tr (x ( χ(Trqk /q (bzx)) f q /q q 2 x∈F y∈F z∈F q
qk
=
q
1 X X χ(ay)χ1 (yx ( q 2 x∈F y∈F
χ2 (bzx))
z∈Fq
qk
q
1 X XX χ(ay)χ1 (yx q 2 x∈F y∈F∗ z∈F∗ qk
= q k−2 +
X
q k −1 1 X X 1 X X q f −1 )) + χ(ay)χ (yx χ2 (bzx)) ( ( 1 q 2 x∈F y∈F∗ q 2 x∈F z∈F∗ qk
+
))(
q
qk
= q k−2 +
q k −1 q f −1
q k −1 q f −1
q
)χ2 (bzx)
q
q
q k −1 q −q 1 X XX q f −1 )χ (bzx). χ(ay)χ (yx + 1 2 q 2 (q f − 1) q 2 x∈F y∈F∗ z∈F∗
k
f
qk
Let S(b) :=
X XX
x∈Fqk y∈F∗q z∈F∗q
q
q
χ(ay)χ1 (yx
q k −1 q f −1
)χ2 (bzx).
LINEAR CODES
15
Then we have Nb = q k−2 +
1 qk − qf + 2 S(b). 2 f q (q − 1) q
(4.2)
In the following, we use Gauss sums to express the exponential sum S(b), b ∈ F∗qk . Lemma 4.2. Let b ∈ F∗qk , f |k and gcd( fk , q − 1) = 1. If f = 1, we have S(b) = −q; if f > 1, then k −1 f
f
S(b) =
q (1 − q) (−1) (q − 1)q − f f q −1 q −1
f
q f −1 −1 q−1
X
k
q k −1
ϕj (−1)G(ϕj , χ1 ) f −1 ϕ¯j (b qf −1 ),
j=1
where ϕ is a multiplicative character of order
q f −1 q−1
of Fqf .
Proof. By using the method computing the exponential sum ∆(b) in the proof of Lemma 3.5, we can similarly obtain that k q k −1 k (−1) f −1 X G(ψ1 , χ1 ) f G(ψ¯1 , χ1 )ψ¯1 (b qf −1 ) S(b) = (1 − q) + f q −1 ∗
b ψ1 ∈F
X
χ(ay)ψ1 (y)
X
qf
k ψ¯1 (z f )
z∈F∗q
y∈F∗q
Since the norm function Nqf /q : F∗qf → F∗q , x 7→ z = x gcd( fk , q − 1) = 1, we have X
k ψ¯1 (z f ) =
X
q f −1 q−1
, is an epimorphism and
ψ¯1 (z)
z∈F∗q
z∈F∗q
=
f −1 q − 1 X ¯ qq−1 ψ (x ). 1 q f − 1 x∈F∗ qf
Note that X
ψ¯1 (x
q f −1 q−1
)=
x∈F∗f q
(
f
−1 q f − 1, if ord(ψ1 )| qq−1 , 0, otherwise.
It is easy to deduce that X
q −1 q −1 q−1 X q−1 q−1 )ψ (x ) χ(ax 1 1 1 q f − 1 x ∈F∗ f
χ(ay)ψ1 (y) =
y∈F∗q
1
qf
and X
x1 ∈F∗f q
q f −1
χ(ax1q−1 ) = −
qf − 1 . q−1
f
16
Z. HENG AND Q. YUE
Hence, we have k −1 f
S(b) = (1 − q) + X
(q − 1) (−1) f (q − 1)2 q f −1 q−1
χ(ax1
2
q f −1 −1 q−1
X
q k −1
k
G(ϕj , χ1 ) f G(ϕ¯j , χ1 )ϕ¯j (b qf −1 )
j=0
)
x1 ∈F∗f q q f −1 −1 q−1
k −1 f
= (1 − q) −
q k −1 k (q − 1) X (−1) j j qf −1 j f G(ϕ ¯ , χ ) ϕ ¯ (b ), G(ϕ , χ ) 1 1 qf − 1 j=0
where ϕ is a multiplicative character of order q f (1−q) q f −1
q f −1 q−1
of Fqf . If f = 1, we have S(b) =
= −q. If f > 1, we have k −1 f
f
q (1 − q) (−1) (q − 1)q S(b) = f − f q −1 q −1
f
q f −1 −1 q−1
X
j
j
ϕ (−1)G(ϕ , χ1 )
k −1 f
j
ϕ¯ (b
q k −1 q f −1
),
j=1
where ϕ is a multiplicative character of order
q f −1 q−1
of Fqf .
By Lemma 4.2, we can determine the value distribution of the exponential sum S(b), b ∈ F∗q , if f = 2. Lemma 4.3. Let b ∈ F∗qk , f |k and gcd( fk , q − 1) = 1. If f = 2, then the value distribution of S(b) is k k q k −1 −q2 +(−1) 2 q 2 +2 , times, q+1 q+1 S(b) = k −1 k +1 2 k −q +(−1) 2 q 2 , q(q −1) times. q+1
q+1
Proof. The proof is similar to that of Lemma 3.7, we omit it here.
Theorem 4.4. Let f |k and gcd( fk , q − 1) = 1. Let CD be the linear code defined in (1.1) with a ∈ F∗q . k −1 If f = 1, then CD is an optimal [ qq−1 , k, q k−1] linear code achieving the Griesmer bound and its weight distribution is given in Table V. If f > 1, then CD is a (q − 1)q f +k−2 − (q f − q)q q f −1 (q k − 1) , k, d ≥ [ qf − 1 qf − 1
k+f −2 2
]
linear code and the Hamming weight w(cb ) of a codeword
cb = (Trqk /q (bd1 ), · · · , Trqk /q (bdn )) ∈ CD , b ∈ F∗qk , equals to f +k−2
w(cb ) =
(q − 1)q qf − 1
+
(−1)
k −1 f
(q − 1)q qf − 1
f −2
q f −1 −1 q−1
X j=1
k
q k −1
ϕj (−1)G(ϕj , χ1 ) f −1 ϕ¯j (b qf −1 ),
LINEAR CODES
17
q f −1 q−1
of Fqf . In particular, if f = 2 and
where ϕ is a multiplicative character of order k ≡ 0 (mod 4), CD is a [
q(q k −1) q 2 −1
, k,
k q k −q 2
q+1
] two-weight linear code and its weight distrik
−1) , k, q bution is given in Table VI; if f = 2 and k ≡ 2 (mod 4), CD is a [ q(qq2 −1 two-weight linear code and its weight distribution is given in Table VII.
k k −q 2 −1
q+1
]
Table V. Weight distribution of CD in Theorem 4.4 for f = 1 weight Frequency 0 1 k−1 k q q −1 Table VI. Weight distribution of CD in Theorem 4.4 for f = 2 and k ≡ 0 (mod 4) weight Frequency 0 1 k
q k −q 2 q+1 k q k +q 2 −1 q+1
q k −1 q+1 q(q k −1) q+1
Table VII. Weight distribution of CD in Theorem 4.4 for f = 2 and k ≡ 2 (mod 4) weight Frequency 0 1 k
q k +q 2 q+1 k q k −q 2 −1 q+1
q k −1 q+1 q(q k −1) q+1
Proof. From Equations (4.1), (4.2) and Lemma 4.2, we can easily obtain the weight distribution of the one-weight linear code CD if f = 1. Since k−1 X qk − 1 q k−1 , ⌈ i ⌉ = q k−1 + q k−2 + · · · + 1 = q q−1 i=0
CD is optimal with respect to the Griesmer bound. Now we assume that f > 1. For a codeword cb = (Trqk /q (bd1 ), · · · , Trqk /q (bdn )) ∈ CD , b ∈ F∗qk , the Hamming weight of it equals to w(cb ) = n − Nb . Then by Equations (4.1), (4.2) and Lemma 4.2, we have f +k−2
w(cb ) =
(q − 1)q qf − 1
+
(−1)
k −1 f
(q − 1)q qf − 1
f −2
q f −1 −1 q−1
X j=1
k
q k −1
ϕj (−1)G(ϕj , χ1 ) f −1 ϕ¯j (b qf −1 ),
18
Z. HENG AND Q. YUE
where ϕ is a multiplicative character of order
| ≤ Then
(−1)
(
p
k −1 f
(q − 1)q qf − 1
k
f −2
q f −1 q−1
of Fqf . Note that
q f −1 −1 q−1
X
j
j
ϕ (−1)G(ϕ , χ1 )
k −1 f
j
ϕ¯ (b
q k −1 q f −1
)|
j=1
q f )( f −1) (q − 1)q f −2 q f − 1 ( − 1). qf − 1 q−1
p k (q − 1)q f +k−2 ( q f )( f −1) (q − 1)q f −2 q f − 1 w(cb ) ≥ − ( − 1) qf − 1 qf − 1 q−1 k+f
(q − 1)q f +k−2 − (q f − q)q 2 −2 = >0 qf − 1 for any b ∈ F∗qk . This implies that the dimension of CD is k. In particular, for f = 2, the weight distribution of CD can be obtained by Equations (4.1), (4.2) and Lemma 4.3. Remark 4.5. By Theorem 4.4, we find that the weight of a codeword cb , b ∈ F∗qk , is the same for any a ∈ F∗q . In fact, there exists an element c ∈ F∗qk such that Nqk /qf (c) = c defining set
q k −1 q f −1
= − a1 because the norm function Nqk /qf is a surjection. Hence, the q k −1
D = {x ∈ Fqk : Trqf /q (x qf −1 ) = −a}
1 qk −1 = {x ∈ Fqk : Trqf /q (− x qf −1 ) = 1} a q k −1
= {x ∈ Fqk : Trqf /q ((cx) qf −1 ) = 1} = {x ∈ Fqk : Trqf /q (x
q k −1 q f −1
) = 1}.
For f = 1, we remark that the linear code CD is equivalent to a class of irreducible cyclic codes, which was investigated by [10, Theorem 15]. For f = 2, we remark that the weight distribution of CD is new. Remark 4.6. In Theorem 4.4, for f = k = 2, we obtain that CD is a [q, 2, q − 1] MDS linear code, which is optimal with respect to the Singleton bound. Example 4.7. Let f = 2, k = 4 and q = 2. Then CD is an optimal [10, 4, 4] twoweight linear code achieving the Griesmer bound. Its weight distribution is given by 1 + 5z 4 + 10z 6 . Example 4.8. Let f = 2, k = 6 and q = 2. Then CD is an optimal [42, 6, 20] two-weight linear code achieving the Griesmer bound. Its weight distribution is given by 1 + 42z 20 + 21z 36 .
LINEAR CODES
19
5. concluding remarks In this paper, we have a class of linear codes with only a few weights and determined its weight distribution in some special cases. Some optimal or almost optimal codes are obtained. An application of a linear code C over Fq is constructing secret sharing , then the linear code can be used schemes introduced in [25, 33]. If wmin/wmax > q−1 q to construct secret sharing schemes with interesting access structures [33]. For the code in Theorem 3.8 for f = 2 and k ≡ 0 (mod 4), we have k
wmin q−1 q k−1 − q 2 −1 > = . k k−1 wmax q q + q2
For the code in Theorem 3.8 for f = 2 and k ≡ 2 (mod 4), we have k
wmin q−1 q k−1 − q 2 > = . k wmax q q k−1 + q 2 −1 if k ≥ 6. For the code in Theorem 4.4 for f = 1, we have q−1 wmin =1> . wmax q
For the code in Theorem 4.4 for f = 2 and k ≡ 0 (mod 4), we have k
qk − q 2 wmin q−1 = . > k −1 k wmax q q + q2
For the code in Theorem 4.4 for f = 2 and k ≡ 2 (mod 4), we have k
wmin q−1 q k − q 2 −1 > = , k wmax q qk + q 2 if k > 2. Hence, these linear codes in this paper can be employed in secret sharing schemes using the framework in [33]. To conclude this paper, we present some open problems in the following: (1) Determine the weight distribution of CD defined in (1.1) for f ≥ 3 if a = 0; (2) Determine the weight distribution of CD defined in (1.1) for f ≥ 3 and gcd( fk , q − 1) = 1 if a ∈ F∗q ; (3) Determine the weight distribution of CD defined in (1.1) for gcd( fk , q − 1) ≥ 2 if a ∈ F∗q .
We believe that it could be an interesting work to settle these problems. References
[1] L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Inf. Contr. 20 (2) (1972) 158-175. [2] M. A. D. Boer, Almost MDS codes, Des. Codes Cryptogr. 9 (1996) 143-155. [3] B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi sums, J. Wiley and Sons Company, New York, 1997. [4] C. Carlet, C. Ding, and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory 51 (6) (June 2005) 2089-2102.
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[5] C. Ding, A Construction of Binary Linear Codes from Boolean Functions, arXiv:1511.00321 [cs. IT]. [6] C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory 61 (6) (2015) 3265-3275. [7] C. Ding, J. Luo, H. Niederreiter, Two-weight codes punctured from irreducible cyclic codes, in Proc. 1st Int. Workshop Coding Theory Cryptography, Y. Li, S. Ling, H. Niederreiter, H. Wang, C. Xing, and S. Zhang, Eds., Singapore, (2008) 119-124. [8] C. Ding, H. Niederreiter, Cyclotomic linear codes of order 3, IEEE Trans. Inf. Theory 53 (6) (June 2007) 2274-2277. [9] C. Ding, X. Wang, A coding theory construction of new systematic authentication codes, Theoretical Computer Science 330 (1) (2005) 81-99. [10] C. Ding, J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math. 313 (4) (Feb. 2013) 434-446. [11] C. Ding, Y. Gao, Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory 59 (12) (Dec. 2013) 7940-7946. [12] K. Ding, C. Ding, Binary linear codes with three weights, IEEE Commun. Letters 18 (11) (2014) 1879-1882. [13] K. Ding, C. Ding, A Class of Two-Weight and Three-Weight Codes and Their Applications in Secret Sharing, IEEE Trans. Inf. Theory, 61 (11) (Nov. 2015) 5835-5842. [14] F. De Clerk, M. Delanote, Two-weight codes, partial geometries and Steiner systems, Des. Codes Cryptogr. 21 (2000) 87-98. [15] P. Delsarte, Weights of linear codes and strongly regular normed spaces, Discrete Math. 3 (1972) 47-64. [16] M. Grassl, Bounds on the minimum distance of linear codes, available online at http://www.codetables.de. [17] Z. Heng, Q. Yue, A class of binary linear codes with at most three weights, IEEE Commun. Letters 19 (9) (2015) 1488-1491. [18] Z. Heng, Q. Yue, Several classes of cyclic codes with either optimal three weights or a few weights, arXiv:1510.05355 [cs. IT]. [19] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1982. [20] C. Li, Q. Yue, and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appli. 28 (2014) 94-114. [21] C. Li, Q. Yue, and F. Li, Hamming weights of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory 56 (6) (2014) 2568-2570. [22] C. Li, S. Bae, J. Ahn, et al, Complete weight enumerators of some linear codes and their applications, Des. Codes Crypto., DOI 10.1007/s10623-015-0136-9. [23] R. Lidl, H. Niederreiter, Finite Fields, Cambridge Univ. Press, Cambridge, 1984. [24] F. Li, Q. Wang, D. Lin, A class of three-weight and five-weight linear codes, arXiv:1509.06242 [cs. IT]. [25] A. Shamir, How to share a secret, Commun. Assoc. Comp. Mach. 22 (11) (1979) 612-613. [26] C. Tang, Y. Qi, Two-weight and three-weight linear codes from weakly regular bent functions, arXiv:1507.06148 [cs. IT]. [27] G. Vega, A critical review and some remarks about one and two-weight irreducible cyclic codes, Finite Fields Appli. 33 (2015) 1-13. [28] G. Vega, A characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field, arXiv:1508.05077 [cs. IT].
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[29] Q. Wang, K. Ding, R. Xue, Binary Linear Codes With Two Weights, IEEE Commun. Letters 19 (7) (2015) 1097 - 1100. [30] C. Xiang, Linear codes from a generic construction, Crypto. Commun., DOI 10.1007/s12095015-0158-1. [31] C. Xiang, A Family of Three-Weight Binary Linear Codes, arXiv:1505.07726 [cs. IT]. [32] C. Xiang, K. Feng, C. Tang, A construction of linear codes over F2t from Boolean functions, arXiv:1511.02264v1 [cs. IT]. [33] J. Yuan, C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory 52 (1) (2006) 206-212. [34] Z. Zhou, N. Li, C. Fan, et al., Linear Codes with Two or Three Weights From Quadratic Bent Functions, Des. Codes Cryptogr., DOI 10.1007/s10623-015-0144-9. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, P. R. China E-mail address: [email protected] Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, P. R. China E-mail address: [email protected]
arXiv:1511.02093v2 [cs.IT] 18 Nov 2015
ZILING HENG AND QIN YUE Abstract. Linear codes with a few weights have been widely investigated in recent years. However, most of these known linear codes were defined over a prime field. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of q-ary linear codes, where q is a power of a prime. The lower bound of the minimum Hamming distance of the codes is obtained. And in some special cases, we evaluate the weight distributions of the linear codes based on semi-primitive Gauss sums and obtain some one-weight, two-weight linear codes. It is quite interesting that we find a few optimal codes achieving some certain bounds on linear codes. The linear codes in this paper can be used in secret sharing schemes, authentication codes and data storage systems.
1. Introduction Let Fq denote the finite field with q elements, where q is a power of a prime p. An [n, l, d] linear code C over Fq is a l-dimensional subspace of Fnq with minimum Hamming distance d. There are some bounds on linear codes. Let nq (l, d) be the minimum length n for which an [n, l, d] linear code over Fq exists. The well-known Griesmer bound is given by l−1 X d nq (l, d) ≥ ⌈ i ⌉. q i=0 The Singleton bound is given by
nq (l, d) ≥ l + d − 1. An [n, l, d] code is called optimal if no [n, l, d + 1] code exists, and is called almost optimal if the [n, l, d + 1] code is optimal. Let Ai denote the number of codewords with Hamming weight i in a code C with length n. The weight enumerator of C is defined by 1 + A1 z + · · · + An z n . The sequence (A1 , A2 , · · · , An ) is called the weight distribution of C. The code C is called to be t-weight if the number of nonzero Aj , 1 ≤ j ≤ n, in the sequence 2010 Mathematics Subject Classification. 11T71, 11T55. Key words and phrases. linear codes, weight distribution, secret sharing schemes. The paper is supported by NNSF of China (No. 11171150) Fundamental Research Funds for the Central Universities (No. NZ2015102); Funding of Jiangsu Innovation Program for Graduate Eduction (the Fundamental Research Funds for the Central Universities; No. KYZZ15 0086). 1
2
Z. HENG AND Q. YUE
(A1 , A2 , · · · , An ) equals to t. Weight distribution is an interesting topic and was investigated by [1, 4, 10, 11, 17, 18, 20, 21, 27, 28]. It could be used to estimate the error-correcting capability and the error probability of error detection of a code. Let D = {d1 , d2 , . . . , dn } ⊆ Fr , where r is a power of q. Let Trr/q be the trace function from Fr onto Fq . A linear code of length n over Fq is defined by CD = {(Trr/q (xd1 ), Trr/q (xd2 ), . . . , Trr/q (xdn )) : x ∈ Fr }. The set D is called the defining set of CD . Although different orderings of the elements of D result in different codes CD , these codes are permutation equivalent and have the same length, dimension and weight distribution. Hence, the orderings of the elements of D will not affect the results in this correspondence. If the set D is well chosen, the code CD may have good parameters. This construction is generic in the sense that many known codes [5, 6, 7, 8, 12, 13, 17, 22, 24, 26, 29, 30, 31, 32, 34] could be produced by selecting the defining set. However, most of these known codes focused on linear codes over a prime field. Let Trqk /q , Trqf /q denote the trace functions from Fqk to Fq and Fqf to Fq , respectively. Let f, k be positive integers such that f |k. In this paper, a class of q-ary linear codes CD is defined by CD = {(Trqk /q (xd1 ), Trqk /q (xd2 ), . . . , Trqk /q (xdn )) : x ∈ Fqk } F∗qk
(1.1)
q k −1 q f −1
) + a = 0}, where a ∈ Fq . Let Nqk /qf with the defining set D = {x ∈ : Trqf /q (x be the norm function from Fqk to Fqf . In fact, the defining set D is constructed from q k −1
the composite function Trqf /q ◦Nqk /qf due to Trqf /q (x qf −1 ) = Trqf /q (Nqk /qf (x)). We investigate this class of linear codes in the following cases: (1) a = 0, f > 1; (2) a ∈ F∗q , gcd( fk , q − 1) = 1. We use Gauss sums to represent their Hamming weights and obtain the lower bounds of their minimum distances. For f = 2 in Case (1) and f = 1, 2 in Case (2), the weight distributions of the linear codes are explicitly determined. Some codes with one or two weights are obtained. In particular, we obtain some codes which are optimal or almost optimal with respect to some certain bounds on linear codes. Two-weights codes are closely related to strongly regular graphs, partial geometries and projective sets [14, 15]. Linear codes with a few weights have applications in secret sharing schemes [25, 33] and authentication codes [9]. For convenience, we introduce the following notations in this paper:
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primitive element of Fqk ,
α β=α χ χ1 χ2 ψ ψ1 ψ2
3
q k −1 q f −1
primitive element of Fqf , canonical additive character of Fq , canonical additive character of Fqf , canonical additive character of Fqk , multiplicative character of Fq , multiplicative character of Fqf , multiplicative character of Fqk . 2. Gauss sums
Let Fq be a finite field with q elements, where q is a power of a prime p. The canonical additive character of Fq is defined as follows: Trq/p (x)
χ : Fq −→ C∗ , χ(x) = ζp
,
where ζp denotes the p-th primitive root of unity and Trq/p is the trace function from Fq to Fp . The orthogonal property of additive characters [23] is given by: ( X q, if a = 0, χ(ax) = 0 otherwise. x∈F q
Let ψ : F∗q −→ C∗ be a multiplicative character of F∗q . The trivial multiplicative character ψ0 is defined by ψ0 (x) = 1 for all x ∈ F∗q . It is known [23] that all the b∗ , which is isomorphic to F∗ . multiplicative characters form a multiplication group F q q The orthogonal property of a multiplicative character ψ is given by (see [23]): ( X q − 1, if ψ = ψ0 , ψ(x) = 0 otherwise. x∈F∗ q
The Gauss sum over Fq is defined by G(ψ, χ) =
X
ψ(x)χ(x).
x∈F∗q
¯ χ) = ψ(−1)G(ψ, χ). If ψ 6= ψ0 , we It is easy to see that G(ψ0 , χ) = −1 and G(ψ, √ have |G(ψ, χ)| = q. Gauss sum is an important tool in this paper to compute exponential sums. In general, the explicit determination of Gauss sums is a difficult problem. In some cases, Gauss sums are explicitly determined in [3, 10, 23]. The semi-primitive case Gauss sums, which will be used in this correspondence, are known as follows. Lemma 2.1. [3] Let λ be a multiplicative character of order N of F∗r and ρ the canonical additive character of Fr . Assume that N 6= 2 and their exists a least
4
Z. HENG AND Q. YUE
positive integer j such that pj ≡ −1 (mod N). Let r = p2jγ for some integer γ. Then the Gauss sums of order N over Fr are given by ( √ if p = 2, (−1)γ−1 r, G(λ, ρ) = γ(pj +1) √ γ−1+ N (−1) r, if p ≥ 3. Furthermore, for 1 ≤ s ≤ N − 1, the Gauss sums G(ψ s ) are given by ( √ j (−1)s r, if N is even, p, γ and p N+1 are odd, s G(λ , ρ) = √ (−1)γ−1 r, otherwise. 3. The case a = 0 Let f be a positive integer such that f |k and k > f > 1. In this section, we investigate CD defined in (1.1) with the defining set q k −1
D = {d1 , d2 , · · · , dn } = {x ∈ F∗qk : Trqf /q (x qf −1 ) = 0}. To determine the length n of CD , we need a lemma below. Lemma 3.1. For the canonical additive character χ1 of Fqf , we have X X
x∈Fqk y∈Fq
χ1 (yx
q k −1 q f −1
) = q + qk − 1 −
(q k − 1)(q − 1) . qf − 1
Proof. For 0 ≤ j ≤ q k −2, let j = (q f −1)s+t with 0 ≤ s ≤ Then X X
x∈Fqk y∈Fq
q k −1
χ1 (yx qf −1 ) = q + q k − 1 +
X X
q k −1 −1 q f −1
and 0 ≤ t ≤ q f −2.
q k −1
χ1 (yx qf −1 )
x∈F∗k y∈F∗q q
k
k
= q+q −1+
−2 X qX
= q + qk − 1 +
−1
f
q −2 −1 X qfX X
y∈F∗q
q k −1
χ1 (yα qf −1
((q f −1)s+t)
)
t=0
s=0
qk − 1 X X χ1 (yz) q f − 1 y∈F∗ z∈F∗ q
= q + qk − 1 −
j
y∈F∗q j=0
q k −1
= q + qk − 1 +
q k −1
χ1 (yα qf −1 )
qf
k
(q − 1)(q − 1) . qf − 1
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5
q k −1
Let n0 = |{x ∈ Fqk : Trqf /q (x qf −1 ) = 0}|. Note that χ1 = χ ◦ Trqf /q . Then q k −1 1 X X q χ(y Trqf /q (x f −1 )) = q x∈F y∈F
n0
q
qk
q k −1 1 X X χ1 (yx qf −1 ) q x∈F y∈F
=
q
qk
= 1+ by Lemma 3.1. Hence,
For each b ∈ F∗qk , let
(q k − 1)(q f − q) q(q f − 1)
(q k − 1)(q f − q) n = n0 − 1 = . q(q f − 1)
(3.1)
q k −1
Nb = |{x ∈ Fqk : Trqf /q (x qf −1 ) = 0 and Trqk /q (bx) = 0}|. By the basic facts of additive characters, for each b ∈ F∗qk we have Nb
X q k −1 1 X X q = 2 χ(Trqf /q (yx f −1 ))( ( χ(Trqk /q (bzx)) q x∈F y∈F z∈F q
qk
=
q
X q k −1 1 X X q f −1 ))( ( χ (yx χ2 (bzx)) 1 q 2 x∈F y∈F z∈F q
qk
= q k−2 +
q
1 X X 1 X X q f −1 )) + χ (yx χ2 (bzx)) ( ( 1 q 2 x∈F y∈F∗ q 2 x∈F z∈F∗ q k −1
qk
qk
q
q
q k −1 1 X XX χ1 (yx qf −1 )χ2 (bzx). + 2 q x∈F y∈F∗ z∈F∗ qk
q
q
By Lemma 3.1, we have X X X X q k −1 q k −1 χ1 (yx qf −1 )) ( χ1 (yx qf −1 )) = −q k + ( x∈Fqk y∈Fq
x∈Fqk y∈F∗q
=
(q − 1)(q f − q k ) . qf − 1
By the orthogonal relation of additive characters, we have X X X X χ2 (bzx) = 0. χ2 (bzx)) = ( x∈Fqk z∈F∗q
Let Ω(b) :=
P P
q k −1
z∈F∗q x∈Fqk
χ1 (yx qf −1 )χ2 (bx) and ∆(b) :=
x∈Fqk y∈F∗q
Nb = q k−2 +
P
Ω(bz). Then we have
z∈F∗q
(q − 1)(q f − q k ) 1 + 2 ∆(b). 2 f q (q − 1) q
(3.2)
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Z. HENG AND Q. YUE
In the following, we compute the exponential sum ∆(b), b ∈ F∗qk . Lemma 3.2. [23] Let r be a power of a prime p. Let ρ be a nontrivial additive character of Fr , m ∈ N, and λ a multiplicative character of Fr of order s = gcd(m, r− 1). Then s−1 X X m ¯ j (a0 )G(λj , ρ) ρ(a0 c + a1 ) = ρ(a1 ) λ j=1
c∈Fr
for any a0 , a1 ∈ Fr and a0 6= 0.
Lemma 3.3. [23] (Davenport-Hasse Theorem) Let r be a power of a prime p. Let ρ be an additive and λ a multiplicative character of Fr , not both of them trivial. Suppose ρ and λ are lifted to characters ρ′ and λ′ , respectively, of the extension field E of Fr with [E : Fr ] = t. Then G(λ′ , ρ′ ) = (−1)t−1 G(λ, ρ)t . Lemma 3.4. Let χ1 , χ2 be the canonical additive characters of Fqf and Fqk , respectively. Let f be a positive integer such that f |k. Then for b ∈ F∗qk , X
χ2 (bxq
f −1
k
) = (−1) f −1
x∈F∗k
X
q k −1
k G(ψ1 , χ1 ) f ψ¯1 (b qf −1 ).
b∗ ψ1 ∈F f
q
q
Proof. By Lemma 3.2, we have X X f f χ2 (bxq −1 ) χ2 (bxq −1 ) = −1 + x∈Fqk
x∈F∗k q
q f −2
= −1 + q f −2
X
=
X
ψ¯2j (b)G(ψ2j , χ2 )
j=1
ψ¯2j (b)G(ψ2j , χ2 ),
j=0
where ψ2 is a multiplicative character of order q f − 1 of Fqk . Let Nqk /qf be the norm mapping from Fqk to Fqf . Then ψ2 = ψ1 ◦ Nqk /qf can be seen as the lift of ψ1 from b∗f to F b ∗k . Note that Nqk /qf is an epimorphism. Then ord(ψ1 ) = ord(ψ2 ) = q f − 1. F q q Therefore, by Lemma 3.3, X X f G(ψ1 ◦ Nqk /qf , χ2 )ψ¯1 (Nqk /qf (b)) χ2 (bxq −1 ) = x∈F∗k q
b∗ ψ1 ∈F f q
k
= (−1) f −1
X
q k −1
k G(ψ1 , χ1 ) f ψ¯1 (b qf −1 ).
b∗ ψ1 ∈F f q
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7
Lemma 3.5. Let f be a positive integer such that f > 1 and f |k, then f
2
k −1 f
f
q (q − 1) (−1) q (q − 1) + ∆(b) = f f q −1 q −1
2
q f −1 −1 q−1
X
where ϕ is a multiplicative character of order
j
j
ϕ (−1)G(ϕ , χ1 )
k −1 f
j
ϕ¯ (b
q k −1 q f −1
),
j=1
q f −1 q−1
of Fqf . q k −1
Proof. Let F∗qk = hαi and F∗qf = hβi. Then β = α qf −1 . And there is a coset decomposition as follows: q f −2
F∗qk
[
=
j=0
αj hαq
f −1
i.
Hence, Ω(b) = q − 1 +
X X
q k −1
χ1 (yx qf −1 )χ2 (bx)
x∈F∗k y∈F∗q q
f
= q−1+
−2 X qX
χ1 (yβ j )
y∈F∗q j=0
X
ω∈hαq
f −1
χ2 (bωαj ) i
q f −2
= q−1+
X 1 XX f j χ2 (bxq −1 αj ) χ (yβ ) 1 f q − 1 y∈F∗ j=0 x∈F∗ q
qk
By Lemma 3.4, k X q k −1 k (−1) f −1 X X Ω(b) = (q − 1) + f G(ψ1 , χ1 ) f ψ¯1 (b qf −1 x) χ1 (yx) q − 1 x∈F∗ y∈F∗ ∗ qf
b ψ1 ∈F
q
qf
k X q k −1 k (−1) f −1 X X = (q − 1) + f G(ψ1 , χ1 ) f ψ¯1 (b qf −1 y −1) ψ¯1 (yx)χ1 (yx) q −1 ∗ y∈F∗ x∈F∗
b ψ1 ∈F
qf
q
qf
k X q k −1 k (−1) f −1 X G(ψ1 , χ1 ) f G(ψ¯1 , χ1 )ψ¯1 (b qf −1 ) ψ1 (y). = (q − 1) + f q −1 ∗ ∗ y∈F
b ψ1 ∈F
qf
q
This implies that X Ω(bz) ∆(b) = z∈F∗q
k X X q k −1 k k (−1) f −1 X ¯ ¯ f q = (q − 1) + f ψ¯1 (z f ). ψ1 (y) G(ψ1 , χ1 ) G(ψ1 , χ1 )ψ1 (b f −1 ) q −1 ∗ z∈F∗ y∈F∗
2
b ψ1 ∈F
qf
q
q
8
Z. HENG AND Q. YUE q f −1
Since the norm function Nqf /q : F∗qf → F∗q , x 7→ y = x q−1 , is an epimorphism, X
q f −1 q−1 X q−1 ). ψ (x 1 q f − 1 x∈F∗
ψ1 (y) =
y∈F∗q
Similarly, we have
qf
X
−1 k k ·f q − 1 X ¯ qq−1 ψ¯1 (z f ) = f ). ψ1 (x1 q − 1 ∗ ∗ z∈F x ∈F f
1
q
qf
Then we have k q k −1 k (−1) f −1 (q − 1)2 X ¯ ¯ f G(ψ q f −1 ) ∆(b) = (q − 1) + G(ψ , χ ) , χ ) ψ (b 1 1 1 1 1 (q f − 1)3 ∗
2
b ψ1 ∈F
X
ψ1 (x
q f −1 q−1
)
X
q f −1 k · q−1 f
ψ¯1 (x1
qf
).
x1 ∈F∗f q
x∈F∗f q
Note that X
ψ1 (x
q f −1 q−1
)=
(
−1 q f − 1, if ord(ψ1 )| qq−1 , 0, otherwise,
)=
(
−1 k q f − 1, if ord(ψ1 )| qq−1 · f, 0, otherwise.
x∈F∗f q
and X
q f −1 k · q−1 f
ψ¯1 (x1
x1 ∈F∗f q
f
f
Hence, we have 2
∆(b) = (q − 1) + f
2
(−1)
k −1 f
(q − 1) f q −1
2
q f −1 −1 q−1
k −1 f
(−1) q (q − 1) (q − 1) + = qf − 1 qf − 1 f
=
k −1 f
q (q − 1) (q − 1) (−1) + qf − 1 qf − 1 f
=
2
2
k −1 f
f
X
k f
j
j
j
G(ϕ , χ1 ) G(ϕ¯ , χ1 )ϕ¯ (b
q k −1 q f −1
)
j=0
2
q f −1 −1 q−1
X
G(ϕ , χ1 ) G(ϕ¯ , χ1 )ϕ¯ (b
X
G(ϕ , χ1 ) f ϕj (−1)G(ϕj , χ1 )ϕ¯j (b qf −1 )
j
k f
j
k
j
j
q k −1 q f −1
)
j=1
2
q f −1 −1 q−1
q (q − 1) q (q − 1) (−1) + f f q −1 q −1
q k −1
j=1
2
q f −1 −1 q−1
X
j
k
q k −1
ϕj (−1)G(ϕ , χ1 ) f −1 ϕ¯j (b qf −1 ),
j=1
where ϕ is a multiplicative character of order
q f −1 q−1
of Fqf .
In general, determining the value distribution of ∆(b) is a very difficult problem. In the following, we determine the value distribution of ∆(b) if f = 2.
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9
Lemma 3.6. For an odd integer q, let ζq+1 be the primitive q + 1-th root of complex , we have unity. Then for any integer 1 ≤ s ≤ q and s 6= q+1 2 qs s 3s 5s ζq+1 + ζq+1 + ζq+1 + · · · + ζq+1 = 0,
and (q−1)s
2s 4s 6s ζq+1 + ζq+1 + ζq+1 + · · · + ζq+1
= −1.
qs s 3s 5s Proof. It is clear that {ζq+1 , ζq+1 , ζq+1 , · · · , ζq+1 } is a geometric progression. Hence qs s 3s 5s ζq+1 + ζq+1 + ζq+1 + · · · + ζq+1 2s· q+1
s ζq+1 (1 − ζq+1 2 ) = = 0. 2s 1 − ζq+1
Similarly, (q−1)s
2s 4s 6s ζq+1 + ζq+1 + ζq+1 + · · · + ζq+1 (q−1)s
2s 2s ζq+1 − ζq+1 ζq+1 = −1. = 2s 1 − ζq+1
Lemma 3.7. If f = 2 and f |k, then the value distribution of ∆(b), b ∈ F∗qk , is given as follows: (1) If p = 2, then k k (q−1)(q2 +(−1) 2 −1 q 2 +2 ) , qk −1 times, q+1 q+1 ∆(b) = k k q(q k −1) (q−1)(q2 +(−1) 2 q 2 +1 ) , times. q+1
(2) If p > 2, then q2 (q−1) q+1 k −1 2q+1 (1 + 2 q 2 + k q (q−1) ∆(b) = (1 − q+1 q 2 −1 + q+1 2 k q2 (q−1) (1 − (−q) 2 −1 ), q+1
q+1
k q−1 (−q) 2 −1 ), 2 k q−1 (−q) 2 −1 ), 2
q k −1 times, q+1 q k −1 times, q+1 (q k −1)(q−1) times. q+1
f
−1 Proof. Let f = 2, then qq−1 = q + 1. For the multiplicative character ϕ of order e N = q + 1 = p + 1, G(ϕ, χ1 ) in Lemma 3.5 is just a semi-primitive case Gauss sum over Fq2 by Lemma 2.2. Note that ϕ(−1) = 1 if f = 2. And by the proof of Lemma 3.5, we have k q q k −1 k (−1) 2 −1 (q − 1)2 X j j j q2 −1 2 G(ϕ G(ϕ , χ ) ¯ , χ ) ϕ ¯ (b ) ∆(b) = (q − 1) + 1 1 q2 − 1 j=0
2
(q+1,q 2 )
with ord(ϕ) = q + 1. Let Cs = β s hβ q+1 i, s = 0, 1, · · · , q, be the cyclotomic ¯ = classes of order q + 1 over Fq2 . Without loss of generality, we assume that ϕ(β)
10
Z. HENG AND Q. YUE q k −1
q k −1
(q+1,q 2 )
sj , j = 0, 1, · · · , q, when b q−1 ∈ Cs ζq+1 . It is clear that ϕ¯j (b q−1 ) = ζq+1 q k −1 q−1
0, 1, · · · , q. Choose bs ∈ F∗qk , such that bs ts =
q X
k 2
j
j
(q+1,q 2 )
∈ Cs j
q k −1 q 2 −1
G(ϕ , χ1 ) G(ϕ¯ , χ1 )ϕ¯ (bs
j=0
Hence, ∆(bs ) = (q − 1)2 +
,s =
, s = 0, 1, · · · , q. Denote
), s = 0, 1, · · · , q.
k
(−1) 2 −1 (q−1)2 ts , q 2 −1
q k −1
s = 0, 1, · · · , q. For a fixed 0 ≤ s ≤ q, the
value of ∆(bs ) occurs q+1 times when b runs through F∗qk . (1) Assume that p = 2. Then q is even. Let ϕ¯0 (β 0 ) ϕ¯1 (β 0 ) ϕ¯2 (β 0 ) . . . ϕ¯q−1 (β 0 ) ϕ¯q (β 0 ) ϕ¯0 (β) ϕ¯1 (β) ϕ¯2 (β) . . . ϕ¯q−1 (β) ϕ¯q (β) ϕ¯0 (β 2 ) ϕ¯1 (β 2 ) ϕ¯2 (β 2 ) . . . ϕ¯q−1 (β 2 ) ϕ¯q (β 2 ) T := .. .. .. .. .. .. . . . . . . 0 q−1 1 q−1 2 q−1 q−1 q−1 q ϕ¯ (β ) ϕ¯ (β ) ϕ¯ (β ) . . . ϕ¯ (β ) ϕ¯ (β q−1 ) ϕ¯0 (β q ) ϕ¯1 (β q ) ϕ¯2 (β q ) . . . ϕ¯q−1 (β q ) ϕ¯q (β q ) 1 1 1 ... 1 1 q−1 q 2 1 ζq+1 ζq+1 . . . ζq+1 ζq+1 2(q−1) 2q 4 1 ζ2 ζ . . . ζ ζ q+1 q+1 q+1 q+1 = .. .. .. .. .. .. . . . . . . 1 ζ q−1 ζ 2(q−1) . . . ζ (q−1)2 ζ q(q−1) q+1 q+1 q+1 q+1 (q−1)q q 2q q2 1 ζq+1 ζq+1 . . . ζq+1 ζq+1
(q+1)×(q+1)
(q+1)×(q+1)
which is called the character matrix of Fq2 . And by Lemma 2.5, G(ϕs , χ1 ) = q, 1 ≤ s ≤ q. Hence, k k G(ϕ0 , χ1 ) 2 G(ϕ¯0 , χ1 ) (−1) 2 +1 t0 k 2 k2 −1 G(ϕ, χ1 ) 2 G(ϕ, ¯ χ1 ) q ·q t1 .. .. = T = ... . T . . k 2 k −1 G(ϕq−1 , χ1 ) 2 G(ϕ¯q−1 , χ1 ) q · q 2 tq−1 k k tq q 2 · q 2 −1 G(ϕq , χ1 ) 2 G(ϕ¯q , χ1 ) Note that for 1 ≤ s ≤ q,
qs s 2s ζq+1 + ζq+1 + · · · + ζq+1 = −1.
Hence, we have (
k
k
t0 = (−1) 2 +1 + q 2 +2 , k k ts = (−1) 2 +1 − q 2 +1 , 1 ≤ s ≤ q.
Then the value distribution of ∆(b), b ∈ F∗qk , follows.
LINEAR CODES
(2) Let p > 2. Then q is odd. Let ϕ¯0 (β 0 ) ϕ¯1 (β 0 ) ϕ¯2 (β 0 ) . . . ϕ¯q−1 (β 0 ) ϕ¯q (β 0 ) ϕ¯0 (β) ϕ¯1 (β) ϕ¯2 (β) . . . ϕ¯q−1 (β) ϕ¯q (β) ϕ¯1 (β 2 ) ϕ¯2 (β 2 ) . . . ϕ¯q−1 (β 2 ) ϕ¯q (β 2 ) ϕ¯0 (β 2 ) .. .. .. .. .. .. . . . . . . ′ T := 0 q+1 q+1 q+1 q+1 q+1 ϕ¯ (β 2 ) ϕ¯1 (β 2 ) ϕ¯2 (β 2 ) . . . ϕ¯q−1 (β 2 ) ϕ¯q (β 2 ) .. .. .. .. .. .. . . . . . . 0 q−1 ϕ¯ (β ) ϕ¯1 (β q−1 ) ϕ¯2 (β q−1 ) . . . ϕ¯q−1 (β q−1 ) ϕ¯q (β q−1 ) ϕ¯0 (β q ) ϕ¯1 (β q ) ϕ¯2 (β q ) . . . ϕ¯q−1 (β q ) ϕ¯q (β q ) 1 1 1 ... 1 1 q−1 q 2 1 ζq+1 ζq+1 . . . ζq+1 ζq+1 2(q−1) 2q 2 4 1 ζq+1 ζ . . . ζ ζ q+1 q+1 q+1 .. .. .. .. .. .. . . . . . . = 1 −1 1 . . . 1 −1 .. .. .. .. .. .. . . . . . . 2 2(q−1) (q−1) q(q−1) q−1 ζ . . . ζ 1 ζ ζ q+1 q+1 q+1 q+1 (q−1)q q 2q q2 1 ζq+1 ζq+1 . . . ζq+1 ζq+1 (q+1)×(q+1)
11
(q+1)×(q+1)
which is called the character matrix of Fq2 . And by Lemma 2.5, G(ϕs , χ1 ) = (−1)s q, 1 ≤ s ≤ q. Hence, k k (−1) 2 +1 G(ϕ0 , χ1 ) 2 G(ϕ¯0 , χ1 ) k k 2 G(ϕ, χ1 ) 2 G(ϕ, ¯ χ1 ) q · (−q) 2 −1 t k k 0 q 2 · q 2 −1 G(ϕ2 , χ1 ) 2 G(ϕ¯2 , χ1 ) t1 k q 2 · (−q) k2 −1 G(ϕ3 , χ1 ) 2 G(ϕ¯3 , χ1 ) .. = T′ T′ = . . . . .. .. q 2 · (−q) k2 −1 tq−1 G(ϕq−2 , χ ) k2 G(ϕ¯q−2 , χ ) 1 1 tq k k q−1 2 q−1 −1 2 2 q ·q G(ϕ , χ1 ) G(ϕ¯ , χ1 ) k k q 2 · (−q) 2 −1 G(ϕq , χ1 ) 2 G(ϕ¯q , χ1 ) By Lemma 3.6, we have k k k k (−1) 2 −1 (q+1)q 2 +1 (q−1)q 2 +1 +1 2 + + , t = (−1) 0 2 2 k k +1 +1 ts = (−1) 2 − q 2 , k k k t q+1 = (−1) k2 +1 + (−1) 2 (q+1)q 2 +1 + (q−1)q 2 +1 , 2 2 2
where 1 ≤ s ≤ q and s 6= follows.
q+1 . 2
Then the value distribution of ∆(b), b ∈ F∗qk ,
Theorem 3.8. Let f |k and k > f > 1. Let CD be the linear code defined by (1.1) with a = 0. The for a codeword cb = (Trqk /q (bd1 ), · · · , Trqk /q (bd1 )) ∈ CD , b ∈ F∗qk , the
12
Z. HENG AND Q. YUE
Hamming weight of it equals to k−2
w(cb ) =
k −1 f
f
f
(q − 1)q (q − q) (−1) q (q − 1) − f 2 f q −1 q (q − 1)
where ϕ is a multiplicative character of order
2
q f −1 −1 q−1
X
q k −1
k
j
ϕj (−1)G(ϕ , χ1 ) f −1 ϕ¯j (b qf −1 ),
j=1
q f −1 q−1
over Fqf . And CD is a
(q k − 1)(q f − q) (q − 1)(q f − q)(q k−2 − q [ , k, d ≥ q(q f − 1) qf − 1
k+f −4 2
)
] k−1
k
k −1 2
−1 , k, (q−1)(q q+1−q ) ] linear code. In particular, if f = 2 and k ≡ 0 (mod 4), CD is a [ qq+1 two-weight code with the weight distribution given in Table I; if f = 2 and k ≡ 2 k 2
k−1
k
−1 (mod 4), CD is a [ qq+1 , k, (q−1)(qq+1 −q ) ] two-weight code with the weight distribution given in Table II.
Table I. Weight distribution of CD for f = 2 and k ≡ 0 (mod 4) weight Frequency 0 1 k
(q−1)(q k−1 +q 2 ) q+1 k (q−1)(q k−1 −q 2 −1 ) q+1
q k −1 q+1 q(q k −1) q+1
Table II. Weight distribution of CD for f = 2 and k ≡ 2 weight Frequency 0 1 k
(q−1)(q k−1 −q 2 ) q+1 k (q−1)(q k−1 +q 2 −1 ) q+1
(mod 4)
q k −1 q+1 q(q k −1) q+1
Proof. For a codeword cb = (Trqk /p (bd1 ), · · · , Trqk /p (bd1 )), b ∈ F∗qk , the Hamming weight of it equals to n0 − Nb . Then by Equations (3.1) and (3.2), Lemma 3.5, we have k−2
w(cb ) =
k −1 f
f
f
q (q − 1) (q − 1)q (q − q) (−1) − qf − 1 q 2 (q f − 1)
where ϕ is a multiplicative character of order k −1 f
| ≤ Then we have
f
(−1) q (q − 1) 2 f q (q − 1)
2
2
q f −1 −1 q−1
q f −1 q−1
X j=1
over Fqf . Note that
q f −1 −1 q−1
X
k
j
k
j
q k −1
ϕj (−1)G(ϕ , χ1 ) f −1 ϕ¯j (b qf −1 )|
j=1
p k −1 q f (q − 1)2 q f − 1 ( − 1)( qf ) f q 2 (q f − 1) q − 1 (q − 1)(q f − q)(q k−2 − q w(cb ) ≥ qf − 1
q k −1
ϕj (−1)G(ϕ , χ1 ) f −1 ϕ¯j (b qf −1 ),
k+f −4 2
)
>0
LINEAR CODES
13
due to k > f > 1. Then the dimension of CD is k. For f = 2, the weight distributions can be obtained by Equations (3.1) and (3.2), Lemma 3.7. Example 3.9. Let f = 2, k = 4 and q = 4, CD in Theorem 3.8 is a [51, 4, 36] linear code, which has the same parameters as the best known linear codes according to [16], with weight enumerator 1 + 204z 36 + 51z 48 . This coincides with the result given in Theorem 3.8. Example 3.10. Let f = 2, k = 4 and and q = 9, CD in Theorem 3.8 is a [656, 4, 576] linear code with weight enumerator 1 + 5904z 576 + 656z 648 . This coincides with the result given in Theorem 3.8. It is observed that the weights of CD in Tables I-II have a common divisor q − 1. This indicates that the code CD can be punctured into a shorter code CDe as follows. q k −1
q k −1
k
q k −1
Note that Trqf /q ((ax) qf −1 ) = a f Trqf /q (x q−1 ) = 0 for all a ∈ Fq if Trqf /q (x qf −1 ) = 0. Hence, the defining set of CD defined by (3.1) can be expressed as e = {ade : a ∈ F∗ , de ∈ D}, e D = F∗q D q
(3.3)
e Then by Theorem where dei /dej 6∈ F∗q for every pair of distinct elements dei , dej in D. 3.8, we have the following result. Corollary 3.11. Let f |k and k > f = 2. Let CDe be the linear code with defining set e defined in (3.3). Then C e is a [ qk2 −1 , k] linear code with the weight distributions D D q −1 given in Tables III, IV for k ≡ 0 (mod 4) and k ≡ 2 (mod 4), respectively. Table III. Weight distribution of CDe for f = 2 and k ≡ 0 (mod 4) weight Frequency 0 1 k
q k−1 +q 2 q+1 k q k−1 −q 2 −1 q+1
q k −1 q+1 q(q k −1) q+1
Table IV. Weight distribution of CDe for f = 2 and k ≡ 2 (mod 4) weight Frequency 0 1 k
q k−1 −q 2 q+1 k q k−1 +q 2 −1 q+1
q k −1 q+1 q(q k −1) q+1
Remark 3.12. Let f = 2, k = 4, then CDe in Corollary 3.11 is an optimal [q 2 + 1, 4, q 2 − q] linear code achieving the Griesmer bound.
14
Z. HENG AND Q. YUE
4. The case a ∈ F∗q In this section, we assume that f is a positive integer such that f |k and gcd( fk , q − 1) = 1. Let other notations be the same as that of Section 3. Now we investigate the linear code defined in (1.1) with the defining set q k −1
D = {d1 , d2, · · · , dn } = {x ∈ Fqk : Trqf /q (x qf −1 ) + a = 0}, where a ∈ F∗q . The length of CD equals to n = |D|. We need a lemma given below to obtain the value of n. Lemma 4.1. For the canonical additive character χ of Fq , we have X X
q k −1
χ(y(Trqf /q (x qf −1 ) + a)) =
x∈Fqk y∈Fq
q f (q k − 1) . qf − 1
Proof. The proof is similar to that of Lemma 3.1. We omit it here.
By Lemma 4.1, we have n=
q k −1 1 X X q f −1 (q k − 1) χ(y(Trqf /q (x qf −1 ) + a)) = . q x∈F y∈F qf − 1
(4.1)
q
qk
For each b ∈ F∗qk , let Nb = |{x ∈ Fqk : Trqf /q (x
q k −1 q f −1
) + a = 0 and Trqk /q (bx) = 0}|.
By the basic facts of additive characters and Lemma 4.1, for any b ∈ F∗qk we have Nb =
X q k −1 1 X X q f −1 ) + a)))( χ(y(Tr (x ( χ(Trqk /q (bzx)) f q /q q 2 x∈F y∈F z∈F q
qk
=
q
1 X X χ(ay)χ1 (yx ( q 2 x∈F y∈F
χ2 (bzx))
z∈Fq
qk
q
1 X XX χ(ay)χ1 (yx q 2 x∈F y∈F∗ z∈F∗ qk
= q k−2 +
X
q k −1 1 X X 1 X X q f −1 )) + χ(ay)χ (yx χ2 (bzx)) ( ( 1 q 2 x∈F y∈F∗ q 2 x∈F z∈F∗ qk
+
))(
q
qk
= q k−2 +
q k −1 q f −1
q k −1 q f −1
q
)χ2 (bzx)
q
q
q k −1 q −q 1 X XX q f −1 )χ (bzx). χ(ay)χ (yx + 1 2 q 2 (q f − 1) q 2 x∈F y∈F∗ z∈F∗
k
f
qk
Let S(b) :=
X XX
x∈Fqk y∈F∗q z∈F∗q
q
q
χ(ay)χ1 (yx
q k −1 q f −1
)χ2 (bzx).
LINEAR CODES
15
Then we have Nb = q k−2 +
1 qk − qf + 2 S(b). 2 f q (q − 1) q
(4.2)
In the following, we use Gauss sums to express the exponential sum S(b), b ∈ F∗qk . Lemma 4.2. Let b ∈ F∗qk , f |k and gcd( fk , q − 1) = 1. If f = 1, we have S(b) = −q; if f > 1, then k −1 f
f
S(b) =
q (1 − q) (−1) (q − 1)q − f f q −1 q −1
f
q f −1 −1 q−1
X
k
q k −1
ϕj (−1)G(ϕj , χ1 ) f −1 ϕ¯j (b qf −1 ),
j=1
where ϕ is a multiplicative character of order
q f −1 q−1
of Fqf .
Proof. By using the method computing the exponential sum ∆(b) in the proof of Lemma 3.5, we can similarly obtain that k q k −1 k (−1) f −1 X G(ψ1 , χ1 ) f G(ψ¯1 , χ1 )ψ¯1 (b qf −1 ) S(b) = (1 − q) + f q −1 ∗
b ψ1 ∈F
X
χ(ay)ψ1 (y)
X
qf
k ψ¯1 (z f )
z∈F∗q
y∈F∗q
Since the norm function Nqf /q : F∗qf → F∗q , x 7→ z = x gcd( fk , q − 1) = 1, we have X
k ψ¯1 (z f ) =
X
q f −1 q−1
, is an epimorphism and
ψ¯1 (z)
z∈F∗q
z∈F∗q
=
f −1 q − 1 X ¯ qq−1 ψ (x ). 1 q f − 1 x∈F∗ qf
Note that X
ψ¯1 (x
q f −1 q−1
)=
x∈F∗f q
(
f
−1 q f − 1, if ord(ψ1 )| qq−1 , 0, otherwise.
It is easy to deduce that X
q −1 q −1 q−1 X q−1 q−1 )ψ (x ) χ(ax 1 1 1 q f − 1 x ∈F∗ f
χ(ay)ψ1 (y) =
y∈F∗q
1
qf
and X
x1 ∈F∗f q
q f −1
χ(ax1q−1 ) = −
qf − 1 . q−1
f
16
Z. HENG AND Q. YUE
Hence, we have k −1 f
S(b) = (1 − q) + X
(q − 1) (−1) f (q − 1)2 q f −1 q−1
χ(ax1
2
q f −1 −1 q−1
X
q k −1
k
G(ϕj , χ1 ) f G(ϕ¯j , χ1 )ϕ¯j (b qf −1 )
j=0
)
x1 ∈F∗f q q f −1 −1 q−1
k −1 f
= (1 − q) −
q k −1 k (q − 1) X (−1) j j qf −1 j f G(ϕ ¯ , χ ) ϕ ¯ (b ), G(ϕ , χ ) 1 1 qf − 1 j=0
where ϕ is a multiplicative character of order q f (1−q) q f −1
q f −1 q−1
of Fqf . If f = 1, we have S(b) =
= −q. If f > 1, we have k −1 f
f
q (1 − q) (−1) (q − 1)q S(b) = f − f q −1 q −1
f
q f −1 −1 q−1
X
j
j
ϕ (−1)G(ϕ , χ1 )
k −1 f
j
ϕ¯ (b
q k −1 q f −1
),
j=1
where ϕ is a multiplicative character of order
q f −1 q−1
of Fqf .
By Lemma 4.2, we can determine the value distribution of the exponential sum S(b), b ∈ F∗q , if f = 2. Lemma 4.3. Let b ∈ F∗qk , f |k and gcd( fk , q − 1) = 1. If f = 2, then the value distribution of S(b) is k k q k −1 −q2 +(−1) 2 q 2 +2 , times, q+1 q+1 S(b) = k −1 k +1 2 k −q +(−1) 2 q 2 , q(q −1) times. q+1
q+1
Proof. The proof is similar to that of Lemma 3.7, we omit it here.
Theorem 4.4. Let f |k and gcd( fk , q − 1) = 1. Let CD be the linear code defined in (1.1) with a ∈ F∗q . k −1 If f = 1, then CD is an optimal [ qq−1 , k, q k−1] linear code achieving the Griesmer bound and its weight distribution is given in Table V. If f > 1, then CD is a (q − 1)q f +k−2 − (q f − q)q q f −1 (q k − 1) , k, d ≥ [ qf − 1 qf − 1
k+f −2 2
]
linear code and the Hamming weight w(cb ) of a codeword
cb = (Trqk /q (bd1 ), · · · , Trqk /q (bdn )) ∈ CD , b ∈ F∗qk , equals to f +k−2
w(cb ) =
(q − 1)q qf − 1
+
(−1)
k −1 f
(q − 1)q qf − 1
f −2
q f −1 −1 q−1
X j=1
k
q k −1
ϕj (−1)G(ϕj , χ1 ) f −1 ϕ¯j (b qf −1 ),
LINEAR CODES
17
q f −1 q−1
of Fqf . In particular, if f = 2 and
where ϕ is a multiplicative character of order k ≡ 0 (mod 4), CD is a [
q(q k −1) q 2 −1
, k,
k q k −q 2
q+1
] two-weight linear code and its weight distrik
−1) , k, q bution is given in Table VI; if f = 2 and k ≡ 2 (mod 4), CD is a [ q(qq2 −1 two-weight linear code and its weight distribution is given in Table VII.
k k −q 2 −1
q+1
]
Table V. Weight distribution of CD in Theorem 4.4 for f = 1 weight Frequency 0 1 k−1 k q q −1 Table VI. Weight distribution of CD in Theorem 4.4 for f = 2 and k ≡ 0 (mod 4) weight Frequency 0 1 k
q k −q 2 q+1 k q k +q 2 −1 q+1
q k −1 q+1 q(q k −1) q+1
Table VII. Weight distribution of CD in Theorem 4.4 for f = 2 and k ≡ 2 (mod 4) weight Frequency 0 1 k
q k +q 2 q+1 k q k −q 2 −1 q+1
q k −1 q+1 q(q k −1) q+1
Proof. From Equations (4.1), (4.2) and Lemma 4.2, we can easily obtain the weight distribution of the one-weight linear code CD if f = 1. Since k−1 X qk − 1 q k−1 , ⌈ i ⌉ = q k−1 + q k−2 + · · · + 1 = q q−1 i=0
CD is optimal with respect to the Griesmer bound. Now we assume that f > 1. For a codeword cb = (Trqk /q (bd1 ), · · · , Trqk /q (bdn )) ∈ CD , b ∈ F∗qk , the Hamming weight of it equals to w(cb ) = n − Nb . Then by Equations (4.1), (4.2) and Lemma 4.2, we have f +k−2
w(cb ) =
(q − 1)q qf − 1
+
(−1)
k −1 f
(q − 1)q qf − 1
f −2
q f −1 −1 q−1
X j=1
k
q k −1
ϕj (−1)G(ϕj , χ1 ) f −1 ϕ¯j (b qf −1 ),
18
Z. HENG AND Q. YUE
where ϕ is a multiplicative character of order
| ≤ Then
(−1)
(
p
k −1 f
(q − 1)q qf − 1
k
f −2
q f −1 q−1
of Fqf . Note that
q f −1 −1 q−1
X
j
j
ϕ (−1)G(ϕ , χ1 )
k −1 f
j
ϕ¯ (b
q k −1 q f −1
)|
j=1
q f )( f −1) (q − 1)q f −2 q f − 1 ( − 1). qf − 1 q−1
p k (q − 1)q f +k−2 ( q f )( f −1) (q − 1)q f −2 q f − 1 w(cb ) ≥ − ( − 1) qf − 1 qf − 1 q−1 k+f
(q − 1)q f +k−2 − (q f − q)q 2 −2 = >0 qf − 1 for any b ∈ F∗qk . This implies that the dimension of CD is k. In particular, for f = 2, the weight distribution of CD can be obtained by Equations (4.1), (4.2) and Lemma 4.3. Remark 4.5. By Theorem 4.4, we find that the weight of a codeword cb , b ∈ F∗qk , is the same for any a ∈ F∗q . In fact, there exists an element c ∈ F∗qk such that Nqk /qf (c) = c defining set
q k −1 q f −1
= − a1 because the norm function Nqk /qf is a surjection. Hence, the q k −1
D = {x ∈ Fqk : Trqf /q (x qf −1 ) = −a}
1 qk −1 = {x ∈ Fqk : Trqf /q (− x qf −1 ) = 1} a q k −1
= {x ∈ Fqk : Trqf /q ((cx) qf −1 ) = 1} = {x ∈ Fqk : Trqf /q (x
q k −1 q f −1
) = 1}.
For f = 1, we remark that the linear code CD is equivalent to a class of irreducible cyclic codes, which was investigated by [10, Theorem 15]. For f = 2, we remark that the weight distribution of CD is new. Remark 4.6. In Theorem 4.4, for f = k = 2, we obtain that CD is a [q, 2, q − 1] MDS linear code, which is optimal with respect to the Singleton bound. Example 4.7. Let f = 2, k = 4 and q = 2. Then CD is an optimal [10, 4, 4] twoweight linear code achieving the Griesmer bound. Its weight distribution is given by 1 + 5z 4 + 10z 6 . Example 4.8. Let f = 2, k = 6 and q = 2. Then CD is an optimal [42, 6, 20] two-weight linear code achieving the Griesmer bound. Its weight distribution is given by 1 + 42z 20 + 21z 36 .
LINEAR CODES
19
5. concluding remarks In this paper, we have a class of linear codes with only a few weights and determined its weight distribution in some special cases. Some optimal or almost optimal codes are obtained. An application of a linear code C over Fq is constructing secret sharing , then the linear code can be used schemes introduced in [25, 33]. If wmin/wmax > q−1 q to construct secret sharing schemes with interesting access structures [33]. For the code in Theorem 3.8 for f = 2 and k ≡ 0 (mod 4), we have k
wmin q−1 q k−1 − q 2 −1 > = . k k−1 wmax q q + q2
For the code in Theorem 3.8 for f = 2 and k ≡ 2 (mod 4), we have k
wmin q−1 q k−1 − q 2 > = . k wmax q q k−1 + q 2 −1 if k ≥ 6. For the code in Theorem 4.4 for f = 1, we have q−1 wmin =1> . wmax q
For the code in Theorem 4.4 for f = 2 and k ≡ 0 (mod 4), we have k
qk − q 2 wmin q−1 = . > k −1 k wmax q q + q2
For the code in Theorem 4.4 for f = 2 and k ≡ 2 (mod 4), we have k
wmin q−1 q k − q 2 −1 > = , k wmax q qk + q 2 if k > 2. Hence, these linear codes in this paper can be employed in secret sharing schemes using the framework in [33]. To conclude this paper, we present some open problems in the following: (1) Determine the weight distribution of CD defined in (1.1) for f ≥ 3 if a = 0; (2) Determine the weight distribution of CD defined in (1.1) for f ≥ 3 and gcd( fk , q − 1) = 1 if a ∈ F∗q ; (3) Determine the weight distribution of CD defined in (1.1) for gcd( fk , q − 1) ≥ 2 if a ∈ F∗q .
We believe that it could be an interesting work to settle these problems. References
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LINEAR CODES
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[29] Q. Wang, K. Ding, R. Xue, Binary Linear Codes With Two Weights, IEEE Commun. Letters 19 (7) (2015) 1097 - 1100. [30] C. Xiang, Linear codes from a generic construction, Crypto. Commun., DOI 10.1007/s12095015-0158-1. [31] C. Xiang, A Family of Three-Weight Binary Linear Codes, arXiv:1505.07726 [cs. IT]. [32] C. Xiang, K. Feng, C. Tang, A construction of linear codes over F2t from Boolean functions, arXiv:1511.02264v1 [cs. IT]. [33] J. Yuan, C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory 52 (1) (2006) 206-212. [34] Z. Zhou, N. Li, C. Fan, et al., Linear Codes with Two or Three Weights From Quadratic Bent Functions, Des. Codes Cryptogr., DOI 10.1007/s10623-015-0144-9. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, P. R. China E-mail address: [email protected] Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, P. R. China E-mail address: [email protected]
