and some new binary codes - Semantic Scholar
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
Quasi-Cyclic Codes Over
and Some New Binary Codes
Nuh Aydin, Student Member, IEEE, and Dwijendra (Dijen) K. Ray-Chaudhuri
Abstract—Recently, (linear) codes over and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study -QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to produces a new binary code, a (92 2 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials. Index Terms—Gray map, quasi-cyclic (QC) codes,
The study of linear codes over 4 and quasi-cyclic (QC) codes over finite fields have provided useful results in coding theory. After the realization of some good binary nonlinear codes from 4 codes in [11], there have been a lot of investigations of the codes over the ring of integers modulo 4. Among the many results in this area, a new binary nonlinear code was constructed with this approach in [3]. On the other hand, many new linear codes have been discovered which are QC [4]–[6], [9], [10], [16]. We combine these two methods and study QC codes over 4 with the goal of finding new binary codes. We have done a computer search to produce some QC codes which improve the known minimum distances for certain lengths and sizes of binary codes. For the purpose of comparison we used Brouwer’s table [2] for linear codes, and Litsyn’s table [13] for nonlinear codes which, in many cases, does not extend to the values of the parameters of our interest. Among the codes constructed is the quaternary [46; 12; 28]-code obtained by lifting the famous binary Golay code which yields a binary (92; 224 ; 28) nonlinear code which is the best among all binary codes. In the process of the search, we also constructed codes with the same parameters as the best known binary linear codes. There are two up-to-date tables of best known linear codes (over the finite fields of order 2; 3; 4; 5; 7; and 9 up to certain lengths and dimensions) and best known binary nonlinear codes (up to minimum distance 30) available on the World Wide Web [2], [13] maintained by Brouwer and Litsyn. We will briefly recall some of the basic definitions and facts for linear codes over 4 . A detailed treatment of this can be found in [18], [14], [11], or [17]. A linear code of length n over 4 (sometimes called a quaternary code, even though this term is also used for codes over the finite field GF (4)) is a 4 -submodule of n 4 . Any 4 linear code C is permutation equivalent to a code with a generator matrix of the form
Ik 0
A
2Ik
where A and C are 2 -matrices and B is a 4 -matrix. Then we say that C is of type 4k 2k and the size of C is 4k 2k . A 4-linear code is not necessarily a free module, a module with a basis. It is so if and only if k2 = 0. In the next section, we give another characterization of this situation for the case of cyclic codes. There are a few types of weight enumerators defined for 4 codes, but we will be mostly interested in Lee weight enumerators. The Lee weights of 0; 1; 2; 3 2 4 are n 0; 1; 2; 1, respectively, and the Lee weight wtL (c ) of a c 2 4 is the rational sum of the Lee weights of its components. This weight function defines a distance dL on n 4 , called the Lee metric. The Lee weight enumerator of a 4 code C is LeeC (x;
codes.
I. INTRODUCTION
G=
2065
B
y) =
c2C
x2n0wt (c) ywt (c) ;
a homogeneous polynomial of degree 2n. The map that is used to obtain binary codes from 4 codes is the Gray map and is defined as follows. First we map 0; 1; 2; 3 to (0; 0); (0; 1); (1; 1) and (1; 0), respectively, then extend it in an 2n obvious way to a map from n 4 to 2 . The Gray image (C ) of a 4 code C of length n will then be a binary code of length 2n. Although the Gray map is not linear and, therefore, the Gray image of a 4 linear code will not be a binary linear code in general, it has the important property that it is an isometry from ( n 4 , Lee distance) to ( 22n , Hamming distance) and Gray images of 4 linear codes are distance-invariant binary codes, even when they are not linear. II. CYCLIC CODES OVER
4
A cyclic code over 4 is a 4 -linear code which is invariant under cyclic shifts. Under the usual identification of vectors with polynomials, cyclic codes of length n are precisely ideals in the ring
R :=
x]
4[
hxn 0 1i :
Some of the most important facts about ideals of this ring and factorization of xn 0 1 are collected below, and they can be found in [18], [14], or [17]. Theorem 1 [14], [18]: Let n be an odd positive integer. Then xn 0 1 can be factored into a product of finitely many pairwise coprime basic irreducible polynomials over 4 , say, xn 0 1 = g1 (x)g2 (x) 1 1 1 gr (x). Also, this factorization is unique up to ordering of the factors. In fact, we have the following: if f2 (x)j(xn 0 1) in 2 [x] then there is a unique monic polynomial f (x) 2 4 [x] such that f (x)j(xn 0 1) in 4 [x] and f (x) = f2 (x), where f (x) denotes the reduction of f (x) modulo 2. The polynomial f (x) in this theorem is called the Hensel lift of f2 (x). One way of finding this polynomial is Graeffe’s method [11], [20] which we illustrate with an example.
2C
Example II.1: Consider the factorization Manuscript received February 21, 2000; revised June 28, 2000. The material in this correspondence was presented in part at the AMS Sectional Meeting, Special Session on Algebraic Coding Theory, Notre Dame University, South Bend, IN, April 2000. N. Aydin was with the Department of Mathematics, The Ohio State University, Columbus, OH 43210 USA. He is now with the Department of Mathematics, Kenyon College, Gambier, OH 43022 USA (e-mail: aydinn@kenyon. edu). D. K. Ray-Chaudhuri is with the Department of Mathematics, The Ohio State University, Columbus, OH 43210 USA (e-mail: dijen@ math.ohio-state.edu). Communicated by P. Solé, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(02)05173-8.
x23 0 1 = (x + 1)(x11 + x10 + x6 + x5 + x4 + x2 + 1) 1(x11 + x9 + x7 + x6 + x5 + x + 1) in
x]. To find the Hensel lift of
2[
f2 (x) = x11 + x10 + x6 + x5 + x4 + x2 + 1 we compute
f (x2 ) := (x11 + x5 )2 0 (x10 + x6 + x4 + x2 + 1)2 mod 4:
0018-9448/02$17.00 © 2002 IEEE
2066
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
power xl p(x);
Then f (x)
=
x
11
is the Hensel lift of are g (x)
=
11
x
10
+ 3x
7
f2 (x)
10
+ 2x
6
5
4
2
+ 2x + x + x + x + x + 2x + 3
. Similarly, Hensel lifts of the other factors 9
7
6
5
4
4 [x].
Theorem 2 [14], [18]: Let n be an odd positive integer and let I be an ideal in R. Then there are unique monic polynomials f (x); g (x); and h(x) over 4 such that I
p(x); xp(x);
=
h
f (x)h(x);
i
2f (x)g (x)
l
...;
x
2
p(x); xp(x); x p(x);
is linearly independent over m < n
Remark 1: The generator polynomial p(x) in the last theorem does not necessarily divide xn 0 1. For example, let n = 3; 3 (x 0 1). f (x) = 1; h(x) = x 0 1 then p(x) = x + 1 and p(x) Remark 2: When h(x) = 1, p(x) = 3f (x) = n x 0 1.
0
f (x)
does divide
We will investigate when p(x) divides xn 0 1 and when it does not. This is related to the code being a free module or not. Recall that a free module is one with a basis. Not every module over a ring need to have a basis. In case the module (over a commutative ring with identity) is free, then every basis of it contains the same number of elements and this number is called the rank of the module.
Definition II.1: Two polynomials f1 (x); f2 (x) 2 4 [x] are said to be relatively prime (or coprime), denoted (f1 (x); f2 (x)) = 1, in 4 [x] if there exist polynomials p1 (x); p2 (x) 2 4 [x] such that p1 (x)f1 (x) + p2 (x)f2 (x) = 1. Theorem II.1: Let I be a cyclic code over 4 of odd length n. Then is a free module of rank k if and only if the generator polynomial p(x) of the corresponding ideal divides xn 0 1 and deg (p(x)) = n 0 k . Proof: (=: This direction is proved in [18]. It is shown that the set fp(x); xp(x); x2 p(x); . . . ; xk01 p(x)g forms a basis for the code. =): Suppose I is free of rank k with a basis fc 1 c 2 ; . . . ; c g. We know that as an ideal I = hp(x)i where p(x) = f (x)h(x) + 2f (x) for some monic polynomials f (x); h(x); g (x) which are pairwise relatively prime with xn 0 1 = f (x)g (x)h(x). Let s = n 0 deg (p(x)). We first show that k = s. Consider the image I of I under reduction mod2. We claim that jI j = 2k . Suppose that there is a relation a1 c1 + a2 c2 + 1 1 1 + ak c = 0 over 2 , where c denotes the reduction of the vector c modulo 2, which is obtained by reducing each component modulo 2. Viewing the same relation over 4 we get a1 c c1 + a2 c 2 + 1 1 1 + ak c = 2c for some 4 -vector c . Multiplying the last equation by 2, and using the fact that fc1 ; c 2 ; . . . ; c g is linearly independent over 4 , we conclude that 2a1 = 2a2 = 1 1 1 = 2ak = 0. Therefore, a1 ; a2 ; . . . ; ak 2 2 4 , hence a1 ; a2 ; . . . ; ak are all 0 in 2 . This proves that fc 1 ; c 2 ; . . . ; c g forms a basis for I over 2, hence jI j = 2k . On the other hand, the image p(x) of p(x) mod 2 divides xn 0 1, which is a generator of I . This implies that xs p(x), and hence any I
4
...;
x
m
g
p(x)
if
0 deg(
p(x))
=
s
=
k:
Therefore, fp(x); xp(x); . . . ; xk01 p(x)g forms a basis for I . Next we are going to show that h(x) = 1, which will imply that n k p(x)j(x 0 1). Consider the vector x p(x). This can be written as
where f (x)g (x)h(x) = xn 0 1 and jI j = 4deg g(x) 2deg h(x) . Theorem 3 [14], [18]: If n is odd, every ideal I of R is principal. More precisely, I = hp(x)i where p(x) = f (x)h(x) + 2f (x) (or f (x)h(x) + 2f (x)g (x)) and f (x); g (x); h(x) are as above.
s
s01
f
+ 3x + 3x + 3x + 3x + 2x + x + 3
and h(x) = x + 3. Moreover, x23 0 1 = f (x)g (x)h(x) in
, can be written as a linear combination of p(x)g. The latter set is also linearly independent, therefore, it is a basis. This shows that jI j = 2s . Hence, k = s. Since p(x) is the generator of I , any c(x) 2 I can be written as c(x) = t(x)p(x) for some t(x) 2 4 [x]. It follows that the set fp(x); xp(x); x2 p(x); . . .g spans I . Moreover, the set
f
k01
k
x p(x)
for some scalars ai n (x 0 1) where
= i=0
2
i
ai x p(x)
4 . This is equivalent to p(x)t(x) = 0 mod
k t(x)
i
= i=0
with n (x
ak
0 1)j
=
1,
ai x
is a monic polynomial of degree k . This means , i.e., f (x)g (x)h(x)jf (x)(2 + h(x))t(x). Since
p(x)t(x)
deg(p(x)) = deg(f (x)) + deg(h(x))
and n
= deg(f (x)) + deg(h(x)) + deg(g (x))
n 0 deg(p(x)) = k . We see that the degrees of both sides of the last divisibility relation are equal. (Note also that deg(h(x)) = n deg(h(x) + 2)). Therefore, x 0 1 = 6f (x)(h(x) + 2)t(x). Now, by the uniqueness of the factorization of xn 0 1 in 4 [x], we conclude that h(x) = 6(h(x) + 2). Hence h(x) = 1.
deg(g (x)) =
Cyclic codes over 4 corresponding to free modules share many of the properties with their counterparts over fields. The following proposition is an example of this which can be proven easily. Proposition II.1: Let C be a cyclic code of odd length m over 4 with a generator p(x) dividing xm 0 1. Then hp(x)i = hp(x)s(x)i if and only if s(x); xp(x0)1 = 1. Note that when the generator polynomial p(x) of a 4 cyclic code of length n divides xn 0 1, then a generator matrix for the code can be taken to be the matrix with the rows p(x); xp(x);
...;
x
n0deg p(x)
p(x):
If, on the other hand, p(x) is of the form p(x)
=
f (x)h(x)
+ 2f (x)g (x)
(with deg h(x) > 0), then a generator matrix for the code is the matrix with the rows f (x)h(x); xf (x)g (x); .
..;x
k
f (x)g (x);
2f (x)g (x); 2xf (x)g (x); . . . ; 2x
where k1
=
n
k2
=
n
and
0 (deg
f (x)
+ deg h(x) + 1)
0 (deg
f (x)
+ deg g (x) + 1):
k
f (x)g (x)
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
III. QUASI-CYCLIC CODES OVER
2067
4
Quasi-cyclic QC and quasi-twisted (QT) codes over fields have been considered by many authors and a large number of new codes of these types have been discovered. We refer the reader to [1], [4]–[7], [9], [10], [12] and [16] for properties of QC codes and the new codes which are QC. In this correspondence, we consider QC codes over the ring 4 , prove some algebraic results, and obtain some new codes. Definition III.1: Let R be a ring. A linear code of length n = ml over R, i.e., an R-submodule of Rn is called an l-QC code over R if it is invariant under cyclic shifts by l positions. An l-QC code of length n = ml over a ring R is an hxR[x0]1i submodule of ( hxR[x0]1i )l . An r -generator QC code is one with a set of r generators. The main facts about 1-generator QC codes over finite fields as given in [12], [15], and [16]. In this work, we only consider 1-generator QC codes over 4 .
Theorem III.1: Let C be a 1-generator l-QC code of length n = ml, with m odd, over 4 with a generator of the form
( ) = ( ( ) 1( )
g x
g x f
( ) ( ) ...
x ; f2 x g x ;
; f
l (x)g(x))
where
[x] example, negacyclic codes are ideals in the ring hx +1 i . Factorization n n of x +1 in 4 [x] is analogous to that of x 0 1 and is, in fact, obtained from it as follows. Let n be an odd integer, and let xn 0 1 = (x 0 1)a(x)b(x) (which is obtained from the factorization of xn 0 1 in 2 [x] by lifting the factors). Applying the transformation (a ring homomorphism) x 70! 0x to this equation, we get xn + 1 = (x + 1)a(0x)b(0x).
Example IV.1: In Example II.1, we obtained the factorization of 0 1 as
23
x
23
x
01=( ( +2 11
x
+ 3 10 + 2 7 + 6 + 5 + 4 + 2 + 2 + 3) 10 + 3 9 + 3 7 + 3 6 + 3 5 + 2 4 + + 3)( + 3)
11
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
:
Then applying the transformation above, we obtain 23
x
+ 1 = ( 11 + 10 + 2 7 + 3 6 + 5 + 3 4 + 3 2 + 2 + 1) ( 11 + 2 10 + 3 9 + 3 7 + 6 + 3 5 + 2 4 + + 1)( + 1) x
x
x
x
x
x
Any ideal in
hx
x
x
x
x
x
x
x
x
x
x
x
x
:
[ ] +1
i
is generated by an element of the form n x + 1 = f (x)g (x)h(x) in 4 [x]. Theorem II.1 can be easily modified to state that a negacyclic code of length n is a free module of rank k if and only if its generator polynomial divides xn + 1 and has degree n 0 k . An l-QT code is a linear code which is invariant under negacyclic shifts by l positions. More information about QT codes can be found in [1]. We can immediately obtain a result similar to Theorem III.1.
( ) = ( ) ( ) + 2 ( ) where
p x
f x h x
f x
( )j( m 0 1) ( ) i ( ) 2 4 [ ] h m 0 1i and ( i ( ) ( )) = 1, ( ) = xg(x0)1 for all 1 . Then C is a free 4 -module of rank 0 deg ( ) and 1 L (C ), where Theorem IV.1: Let C be a 1-generator -QT code of length = , and L (C ) are the minimum Lee weights of the cyclic code generated with odd, over 4 with a generator of the form by ( ) and the code C , respectively. ( ) = ( ( ) 1 ( ) 2 ( ) ( ) . . . l ( ) ( )) Proof: Let 1 . For fixed consider the following th projection map on an -QC code C of length = : m + 1i, and where ( )j( m + 1) ( ) i ( ) 2 4[ ] h 5i : n4 ! m4 ( i ( ) ( )) = 1, ( ) = xg(x+1) for all 1 . Then C is a free 0 deg ( ) and 1 L (C ), where and 4 -module of rank ( 0 1 . . . ml01 ) ! (i01)m 1+(i01)m . . . m01+(i01)m ( C ) are the minimum Lee weights of the negacyclic code generated L Observe that 5i (C ) is a cyclic code generated by h i ( ) ( )i for by ( ) and the code C , respectively. all 1 . We have that one of the components becomes zero if and only if all the others do because ( ) i ( ) ( ) = 0 if and only if ( )j( ( ) i ( )) (if ( ) 6= 0), which implies that ( )j ( ) since V. NEW CODES AND THEIR GENERATORS ( i ( ) ( )) = 1. So, ( ) j ( ) ( ) = 0 for all . Therefore, if is Theorem 5 was the basis of the search method of [16] and [1] where a nonzero codeword in C then 5i ( ) 6= 0 for all . Since h i ( ) ( )i = they many new QC and QT codes. The analogous Theorems III.1 h ( )i, 5i (C ) is a cyclic code with generator polynomial ( ), and (and found IV.1) for 4 codes is the basis of our method of search. We start every nonzero codeword has Lee weight (by assumption). Hence, m 0 1 in 2 [ ] for an odd integer . Then a nonzero codeword in C has a weight greater than or equal to 1 . with a polynomial 2 ( )j m 0 1 in 4 [ ] (or dividing m +1). Moreover, it can be shown, similar to the cyclic code case, that elements we compute its Hensel lift ( )j ( ) ( ) . . . n0(deg(g(x)01) ( ) form a basis for the code. In Note that in this case the cyclic code generated by ( ) is a free module of rank 0 deg ( ). Then we search over 1-generator 4 -QC (or QT) fact, if a relation codes of the form ( ( ) 1 ( ) ( ) 2 ( ) . . . ( ) l ( )) of length deg g (x)01 = . In fact, we always work with the case = 2 in this search. i ( )=0 with i 2 4 i Finally, we map our codes to 2n via the Gray map and get binary g x
f
x
;
x ; h x
g x ; f
x
x = x
h x
i
m
g x
l
d
l
d
d
l
d
g x
i
l
i
l
g x
i
n
ml
g x
f
c ; c ;
; c
i
c
; c
;
f
; c
p x f
x
ml
p x f
x
f
x g x
h x
p x
x ;f
x g x ;
; g x ; f
x ; h x
;f
x g x
x = x
x
h x
i
m
g x
l
d
l
d
d
d
g x
x g x
p x
x ; h x
g x f
:
l
p x f
h x
n
m
x g x
c
j
c
i
f
x g x g x
g x
d
l
g
d
x
x
x
g x
g x ; g x ; xg
g x
g x f
a x g x
;
n
a
g x 01
i
i
( )=0
a x g x
i=0 i bi x g (x) 6= 0, then neither is i
i i bi x g (x)=0.
IV. NEGACYCLIC AND QUASI-TWISTED CODES A 4 -linear code is called negacyclic if it is invariant under the negacyclic shift
(0
c ; c1 ;
...
; c
n01 ) 70! (0cn01 ; c0 ;
...
x ; g x f
x ;
; g x f
ml
x
l
F2
exists (with m-dimensional vectors), then a similar relation deg ( )
x
g x
n
holds in n 4 . Also, if
x
g g x
; x
i=0
m
x
; c
n02 ):
Negacyclic codes over 4 have recently been investigated in [19]. Most of the results about cyclic codes are also true for negacyclic code. For
codes. The parameters of new codes, their generators, and Lee weight enumerators follow. In some cases, we have been able to determine that resulting binary codes are not linear. For each of the codes 1–3, we considered linear span of k vectors (for appropriate k ) and we found words of weights not appearing in the weight enumerators of these codes, showing that these codes are not linear. In general, however, determining linearity of Gray image of a 4 code is not trivial. 1) A (92;
224 28) binary code: Let ( )= 11 +3 10 +2 7 + 6 + 5 + 4 + 2 +2 +3j 23 0 1
g x
;
x
x
x
x
x
x
x
x
x
considered in the Example I.1. Note that this is the lift of the famous binary Golay code to 4 which is studied in details in [8]. And let
( )=2 2 +2 3 + 4 +2 5 +2 7 +2 8 +2 9 +3 10 +
f x
x
x
x
x
x
x
x
x
11
x
:
2068
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
Then the 4 -QC code with the generators (g (x); g (x)f (x)) has length 46, rank 12, and minimum Lee weight 28 (with computer calculations). In fact, its Lee weight enumerator is x
92
64 28
+ 3082x
62 30
+ 10120x
y
58 34
+ 121072x
y
56 36
+ 320068x
52 40
+ 1294164x
y
+ 1969168x + 2538878x + 690368x
y y
34 58
+ 121072x
y
28 64 y
44 48 y
38 54
+ 38111x
+y
54 38
y
y
32 60
92
y
y
+ 690368x
50 42
40 52
+ 1294164x + 3082x
y
46 46
+ 2807152x
60 32
+ 38111x
y
y
48 44
+ 2538878x
y
42 50
+ 1969168x
y
36 56
+ 320068x
y
g2 (x)
6
x
x 01 h (x) ,
=
j
5
+x +1
63
x
+ 10120x
252
132 120
+ 2520x
y
0 1 mod 2
124 128
+ 63x
252
136 116
+1128x
y
128 124
y
+63x
y
126 126
+64x
118 134
+3040x
134 118
+3040x
y
y
y
116 136
+ 1512x
y
:
132 120
+1880x
124 128
+63x
116 132
+1128x
y
+y
y
252
y
120 132
+1880x
y
:
4) A (60; 28 ; 28) binary code: Let h2 (x) = x4 + x + 1jx15 0 1 1 and g2 (x) = xh (0 x) in 2 [x]. Let g (x) be the Hensel lift of g2 (x). Then the 4 -QC code of length 30 generated by 2 3 (g (x); g (x)f (x)) where f (x) = 2x + x has rank 4 and minimum Lee weight 28. Therefore, its Gray image is a 8 (60; 2 ; 28)-code. This code is necessarily nonlinear because its minimum distance exceeds the largest possible minimum distance of a linear code with these parameters [2]. Moreover, it gives the parameters of the best known binary nonlinear code [13]. The Lee weight enumerator of this code is 60
x
32 28
+ 180x
y
28 32
+ 15x
y
32 28
+ 180x
y
24 36
+ 60x
y
30 30
+ 16x
y
y
y
26 34
+ 120x
y
24 36
+ 60x
y
:
6) A (62; 29 ; 28) binary code: This is obtained by extending the previous code by a parity check and has a larger minimum distance than the best known binary linear code with parameters [62; 9; 27]. 7) A (84; 213 ; 34) binary code: Let g (x)
6
5
4
2
3
2
= (x + 3x + 3x + 3x + 2x + 1)(x + 2x + x + 1) 3
2
2
(x + x + 2x + 1)(x + 3x + 1) 21
which divides x
6
f (x)
=
x
h(x)
=
x
+1
in
4 [x],
and let
5
4
2
+ 2x + 3x + 3x + 3x + 1
+1
21
3) A (252; 214 ; 116) binary code: With the same setting and x) the notation as in the last example, let g 0 (x) = gx(+3 . And let 0 4 5 6 f (x) = x + x + 3x . Then the 4 -QC code generated by 0 0 0 (g (x); g (x)f (x)) has length 126, rank 7, and minimum Lee weight 116. Its Gray image, a (252; 214 ; 116) nonlinear code, looks much better than the comparable [252; 14; 112] binary code. The Lee weight enumerator of this code is x
34 26
+ 120x
y
and let g (x) be the Hensel lift of g2 (x) in 4 [x]. Then the 4 -QC code of length 126 generated by 3 4 (g (x); g (x)f (x)) where f (x) = x + 2x has rank 6 and minimum Lee weight 120. Its Gray image is a (252; 212 ; 120) nonlinear code, while the best known binary linear code of length 252 and dimension 12 has minimum distance 118. The upper bound for a binary linear code of this length and dimension is 120 [2]. However, this upper bound does not prove the existence of a code achieving it. The Lee weight enumerator of this code is x
60
x
+15x
Therefore, its Gray image is a (92; 2 ; 28) binary code, which is nonlinear. This code is not only better than the comparable (best known) binary linear code [92; 24; 26] [2], but also establishes a new record among all nonlinear binary codes [13]. In terms of the notation of [13], this improves A(92; 28) = 23 A(91; 27) from 2 to 224 where A(n; d) denotes the size of the largest binary code of length n and minimum distance d. In other words, this code contains twice as many codewords as the previously best known comparable code.
=
1
30 62
24
h2 (x)
C
28 32
:
2) A (252; 212 ; 120) binary code: Let
3 2 = (3x + x )g (x)h(x) + 2g (x)f (x), respectively. Then is free of rank 4 but C2 is not; it is of type 44 21 . Then the 4 -QC code generated by (p1 (x); p2 (x)) has length 30, size 9 2 , and minimum Lee weight 26. Its Gray image gives a binary (60; 29 ; 26)-code, which has the parameters of the best known binary linear code which is also optimal. This code has fewer codewords than the comparable best known nonlinear binary code [13]. The Lee weight enumerator is p2 (x)
:
5) A (60; 29 ; 26) binary code: Let g (x) be as in the last example and let h(x) = x + 3, and f (x) = gx(x)h0(1x) . Consider the 4 cyclic codes C1 and C2 generated by p1 (x) = 3x3 g (x)h(x) and
then x + 1 = f (x)g (x)h(x) and this factorization can be obtained from that of x21 0 1. The polynomial g (x) generates a negacyclic code of length 21 of rank 7. Let g (x )
= (g (x)f1 (x);
g (x)f2 (x))
where 6
5
f1 (x)
= 3x + x
f2 (x)
=
and 6
5
4
3
2
+ 2x + x + x + 2x + 1:
x
The QT code whose generator matrix has the rows g (x ); g (x ); xg
...;
6
x g (x );
2x
7
g (x )
13
has length 42, size 2 , and minimum Lee weight 34. Therefore, its Gray image gives a binary (84; 213 ; 34)-code which has a larger minimum distance than the best known [84; 13; 33] binary code. The Lee weight enumerator of this code is 84
x
50 34
+ 504x
y
42 42
+ 1008x
y
34 50
+ 504x
y
48 36
+ 609x
y
40 44
+ 798x
+y
84
y
46 38
+ 1680x
y
38 46
+ 1680x
y
44 40
+ 798x
y
36 48
+ 609x
y
:
8) A (252; 213 ; 118) binary code: Let f (x) = x6 + x5 + 2x3 + 1, 63 h(x) = x + 1 which both divide x + 1 in 4 [x], and let x +1 g (x) = f (x)h(x) . Let g (x ) = (g (x)f1 (x); g (x)f2 (x)) where 6 5 5 4 f1 (x) = x + x and f2 (x) = x + 3x . Then the 4 QT code with generator matrix having the rows g (x ); g (x) ; xg
...;
6
x g g(x );
2x
7
g (x )
13
has length 126, size 2 , and minimum Lee weight 118. Therefore, its Gray image is a (252; 213 ; 118) binary code with a larger minimum distance than the best known [252; 13; 116] binary linear code. The Lee weight enumerator is as follows: 252
x
134 118
+ 2016x
y
132 120
+ 2520x
124 128
+63x
y
y
126 126
+ 64x
118 134
+ 2016x
y
y
116 136
+ 1512x
y
:
9) A (254; 213 ; 120) binary code: This code is obtained by extending the previous code by a parity check and then taking the Gray image. It has a better minimum distance than the best known [254; 13; 118] binary linear code and has the parameters of an optimal linear code for this length and the size (which is currently hypothetical and yet to be constructed).
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
ACKNOWLEDGMENT The authors wish to thank Prof. J. Ferrar of the Department of Mathematics, The Ohio State University, for his useful discussions and the referees for the suggestions. REFERENCES [1] N. Aydin, I. Siap, and D. K. Ray-Chaudhuri, “The structure of 1-generator quasitwisted codes and new linear codes,” Des., Codes Cryptogr., vol. 23, no. 3, pp. 313–326, December 2001. [2] A. E. Brouwer. Linear code bounds. [Online]. Available: http://www. win.tue.nl/aeb/voorlincod.html [3] A. R. Calderbank and G. McGuire, “Construction of a (64; 2 ; 12) code via Galois rings,” Des. Codes Cryptogr., vol. 10, no. 2, pp. 157–165, 1997. [4] Z. Chen, “Six new binary quasicyclic codes,” IEEE Trans. Inform. Theory, vol. 40, pp. 1666–1667, Sept. 1994. [5] R. N. Daskalov, T. A. Gulliver, and E. Metodieva, “New good quasicyclic ternary and quaternary linear codes,” IEEE Trans. Inform. Theory, vol. 43, pp. 1647–1650, Sept. 1997. , “New ternary linear codes,” IEEE Trans. Inform. Theory, vol. 45, [6] pp. 1687–1688, July 1999. [7] P. P. Greenough and R. Hill, “Optimal ternary quasicyclic codes,” Des. Codes, Cryptogr., vol. 2, pp. 81–91, 1992. [8] M. Greferath and E. Viterbo, “On - and -linear lifts of the Golay code,” IEEE Trans. Inform. Theory, vol. 45, pp. 2524–2527, Nov. 1999. [9] T. A. Gulliver and V. K. Bhargava, “Nine good rate (m 1)=pm quasicyclic codes,” IEEE Trans. Inform. Theory, vol. 38, pp. 1366–1369, July 1992. [10] , “New good rate (m 1)=pm ternary and quaternary quasicyclic codes,” Des., Codes Cryptogr., vol. 7, pp. 223–233, 1996. [11] A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z -linearity of Kerdock, Preperata, Goethals, and related codes,” IEEE Trans. Inform. Theory, vol. 40, pp. 301–319, Mar. 1994. [12] K. Lally and P. Fitzpatrick, “Construction and classification of quasicyclic codes,” in Proc. Workshop on Coding and Cryptography (WCC 99), Paris, France, Jan. 11–14, 1999. [13] S. Litsyn. Table of nonlinear binary codes. [Online]. Available: http://www.eng.tau.ac.il/litysn/tableand/index.html [14] V. S. Pless and Z. Qian, “Cyclic codes and quadratic residue codes over ,” IEEE Trans. Inform. Theory, vol. 42, pp. 1594–1600, Sept. 1996. [15] G. E. Séguin and G. Drolet, “The theory of 1-generator quasicyclic codes,” preprint, 1990. [16] I. Siap, N. Aydin, and D. K. Ray-Chaudhuri, “New ternary quasicyclic codes with better minimum distances,” IEEE Trans. Inform. Theory, vol. 46, pp. 1554–1558, July 2000. [17] J. H. van Lint, Introduction to Coding Theory. Berlin, Germany: Springer-Verlag, 1999. [18] Z. X. Wan, Quaternary Codes. Singapore: World Scientific, 1997. [19] J. Wolfmann, “Negacyclic and cyclic codes over ,” IEEE Trans. Inform. Theory, vol. 45, pp. 2527–2532, Nov. 1999. [20] M. Yamada, “Distance-regular graphs of girth 4 over an extension ring of Z/4Z,” Graphs Comb., vol. 6, pp. 381–394, 1990.
0
0
2069
The -ary Image of Some -ary Cyclic Codes: Permutation Group and Soft-Decision Decoding Jérôme Lacan and Emmanuelle Delpeyroux Abstract—Using a particular construction of generator matrices of the -ary image of -ary cyclic codes, it is proved that some of these codes are invariant under the action of particular permutation groups. The equivalence of such codes with some two-dimensional (2-D) Abelian codes and cyclic codes is deduced from this property. These permutations are also used in the area of the soft-decision decoding of some expanded Reed–Solomon (RS) codes to improve the performance of generalized minimum-distance decoding. Index Terms—Permutation groups, -ary image of soft-decision decoding.
-ary cyclic codes,
I. INTRODUCTION One important area in coding theory is concerned with codes which are invariant under a set of permutations. For example, any linear code of length n over q (i.e., vector subspace of ( q )n ) which has the property of being invariant under the cyclic permutation (shift) is called a cyclic code. This property allows us to consider such a code as a principal ideal in the algebra n q [z ]=(z 0 1). Let (i; j ) denote the greatest common divisor of two integers i and j . When (n; q ) = 1, a cyclic code is entirely defined by the set of its nonzeros, i.e., the nth roots of unity such that at least one codeword polynomial c(x) does not evaluate to zero in this point. The parameters of such a code are usually denoted by [n; k], where k is the dimension of the code considered as a q -vector subspace. Note that k is also the number of nonzeros. In order to distinguish different codes of the same dimension, we denote a code by [n; N Z ]q , where N Z is the set of nonzeros. A generator polynomial g (z ) of such a code is a polynomial codeword such that each codeword c(z ) can be expressed as a product c(z ) = g (z ) 2 u(z ) where u(z ) is a polynomial of degree less than k . From such a generator polynomial, we can obtain a generator matrix G of this code defined as follows:
1 1 1 k01 g(z) where GT denotes the transpose of G. G
T =
g (z ) zg (z )
z
An Abelian code over q is a vector subspace of ( q )mn invariant under the shift of the rows and the columns where we consider each codeword as an m 2 n-matrix [6]. Many researchers have worked on codes which are invariant under particular groups of permutations such as, for example, the general linear group [1]–[3] or the Mathieu group [5, Ch. 20]. The first problem we address in this correspondence is that of determining a group of permutations under whose action the q -ary image of some q m -ary cyclic codes is invariant. This property is used to obtain new results on these codes such as the invariance under particular permutations or the equivalence with cyclic and two-dimensional (2-D) Abelian codes. Permutations that fix a code have an interesting application in the area of soft-decision decoding of the q -ary image (the so-called expanded image) of the Reed–Solomon (RS) codes. For these codes,
Manuscript received September 5, 2000; revised September 7, 2001. J. Lacan is with the Département de Mathématiques Appliquées et d’Informatique, ENSICA, 31056 Toulouse, France (e-mail: [email protected]). E. Delpeyroux is with the Département d’Informatique, ICAM, 31300 Toulouse, France (e-mail: [email protected]). Communicated by R. Koetter, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(02)05155-6. 0018-9448/02$17.00 © 2002 IEEE
Quasi-Cyclic Codes Over
and Some New Binary Codes
Nuh Aydin, Student Member, IEEE, and Dwijendra (Dijen) K. Ray-Chaudhuri
Abstract—Recently, (linear) codes over and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study -QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to produces a new binary code, a (92 2 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials. Index Terms—Gray map, quasi-cyclic (QC) codes,
The study of linear codes over 4 and quasi-cyclic (QC) codes over finite fields have provided useful results in coding theory. After the realization of some good binary nonlinear codes from 4 codes in [11], there have been a lot of investigations of the codes over the ring of integers modulo 4. Among the many results in this area, a new binary nonlinear code was constructed with this approach in [3]. On the other hand, many new linear codes have been discovered which are QC [4]–[6], [9], [10], [16]. We combine these two methods and study QC codes over 4 with the goal of finding new binary codes. We have done a computer search to produce some QC codes which improve the known minimum distances for certain lengths and sizes of binary codes. For the purpose of comparison we used Brouwer’s table [2] for linear codes, and Litsyn’s table [13] for nonlinear codes which, in many cases, does not extend to the values of the parameters of our interest. Among the codes constructed is the quaternary [46; 12; 28]-code obtained by lifting the famous binary Golay code which yields a binary (92; 224 ; 28) nonlinear code which is the best among all binary codes. In the process of the search, we also constructed codes with the same parameters as the best known binary linear codes. There are two up-to-date tables of best known linear codes (over the finite fields of order 2; 3; 4; 5; 7; and 9 up to certain lengths and dimensions) and best known binary nonlinear codes (up to minimum distance 30) available on the World Wide Web [2], [13] maintained by Brouwer and Litsyn. We will briefly recall some of the basic definitions and facts for linear codes over 4 . A detailed treatment of this can be found in [18], [14], [11], or [17]. A linear code of length n over 4 (sometimes called a quaternary code, even though this term is also used for codes over the finite field GF (4)) is a 4 -submodule of n 4 . Any 4 linear code C is permutation equivalent to a code with a generator matrix of the form
Ik 0
A
2Ik
where A and C are 2 -matrices and B is a 4 -matrix. Then we say that C is of type 4k 2k and the size of C is 4k 2k . A 4-linear code is not necessarily a free module, a module with a basis. It is so if and only if k2 = 0. In the next section, we give another characterization of this situation for the case of cyclic codes. There are a few types of weight enumerators defined for 4 codes, but we will be mostly interested in Lee weight enumerators. The Lee weights of 0; 1; 2; 3 2 4 are n 0; 1; 2; 1, respectively, and the Lee weight wtL (c ) of a c 2 4 is the rational sum of the Lee weights of its components. This weight function defines a distance dL on n 4 , called the Lee metric. The Lee weight enumerator of a 4 code C is LeeC (x;
codes.
I. INTRODUCTION
G=
2065
B
y) =
c2C
x2n0wt (c) ywt (c) ;
a homogeneous polynomial of degree 2n. The map that is used to obtain binary codes from 4 codes is the Gray map and is defined as follows. First we map 0; 1; 2; 3 to (0; 0); (0; 1); (1; 1) and (1; 0), respectively, then extend it in an 2n obvious way to a map from n 4 to 2 . The Gray image (C ) of a 4 code C of length n will then be a binary code of length 2n. Although the Gray map is not linear and, therefore, the Gray image of a 4 linear code will not be a binary linear code in general, it has the important property that it is an isometry from ( n 4 , Lee distance) to ( 22n , Hamming distance) and Gray images of 4 linear codes are distance-invariant binary codes, even when they are not linear. II. CYCLIC CODES OVER
4
A cyclic code over 4 is a 4 -linear code which is invariant under cyclic shifts. Under the usual identification of vectors with polynomials, cyclic codes of length n are precisely ideals in the ring
R :=
x]
4[
hxn 0 1i :
Some of the most important facts about ideals of this ring and factorization of xn 0 1 are collected below, and they can be found in [18], [14], or [17]. Theorem 1 [14], [18]: Let n be an odd positive integer. Then xn 0 1 can be factored into a product of finitely many pairwise coprime basic irreducible polynomials over 4 , say, xn 0 1 = g1 (x)g2 (x) 1 1 1 gr (x). Also, this factorization is unique up to ordering of the factors. In fact, we have the following: if f2 (x)j(xn 0 1) in 2 [x] then there is a unique monic polynomial f (x) 2 4 [x] such that f (x)j(xn 0 1) in 4 [x] and f (x) = f2 (x), where f (x) denotes the reduction of f (x) modulo 2. The polynomial f (x) in this theorem is called the Hensel lift of f2 (x). One way of finding this polynomial is Graeffe’s method [11], [20] which we illustrate with an example.
2C
Example II.1: Consider the factorization Manuscript received February 21, 2000; revised June 28, 2000. The material in this correspondence was presented in part at the AMS Sectional Meeting, Special Session on Algebraic Coding Theory, Notre Dame University, South Bend, IN, April 2000. N. Aydin was with the Department of Mathematics, The Ohio State University, Columbus, OH 43210 USA. He is now with the Department of Mathematics, Kenyon College, Gambier, OH 43022 USA (e-mail: aydinn@kenyon. edu). D. K. Ray-Chaudhuri is with the Department of Mathematics, The Ohio State University, Columbus, OH 43210 USA (e-mail: dijen@ math.ohio-state.edu). Communicated by P. Solé, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(02)05173-8.
x23 0 1 = (x + 1)(x11 + x10 + x6 + x5 + x4 + x2 + 1) 1(x11 + x9 + x7 + x6 + x5 + x + 1) in
x]. To find the Hensel lift of
2[
f2 (x) = x11 + x10 + x6 + x5 + x4 + x2 + 1 we compute
f (x2 ) := (x11 + x5 )2 0 (x10 + x6 + x4 + x2 + 1)2 mod 4:
0018-9448/02$17.00 © 2002 IEEE
2066
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
power xl p(x);
Then f (x)
=
x
11
is the Hensel lift of are g (x)
=
11
x
10
+ 3x
7
f2 (x)
10
+ 2x
6
5
4
2
+ 2x + x + x + x + x + 2x + 3
. Similarly, Hensel lifts of the other factors 9
7
6
5
4
4 [x].
Theorem 2 [14], [18]: Let n be an odd positive integer and let I be an ideal in R. Then there are unique monic polynomials f (x); g (x); and h(x) over 4 such that I
p(x); xp(x);
=
h
f (x)h(x);
i
2f (x)g (x)
l
...;
x
2
p(x); xp(x); x p(x);
is linearly independent over m < n
Remark 1: The generator polynomial p(x) in the last theorem does not necessarily divide xn 0 1. For example, let n = 3; 3 (x 0 1). f (x) = 1; h(x) = x 0 1 then p(x) = x + 1 and p(x) Remark 2: When h(x) = 1, p(x) = 3f (x) = n x 0 1.
0
f (x)
does divide
We will investigate when p(x) divides xn 0 1 and when it does not. This is related to the code being a free module or not. Recall that a free module is one with a basis. Not every module over a ring need to have a basis. In case the module (over a commutative ring with identity) is free, then every basis of it contains the same number of elements and this number is called the rank of the module.
Definition II.1: Two polynomials f1 (x); f2 (x) 2 4 [x] are said to be relatively prime (or coprime), denoted (f1 (x); f2 (x)) = 1, in 4 [x] if there exist polynomials p1 (x); p2 (x) 2 4 [x] such that p1 (x)f1 (x) + p2 (x)f2 (x) = 1. Theorem II.1: Let I be a cyclic code over 4 of odd length n. Then is a free module of rank k if and only if the generator polynomial p(x) of the corresponding ideal divides xn 0 1 and deg (p(x)) = n 0 k . Proof: (=: This direction is proved in [18]. It is shown that the set fp(x); xp(x); x2 p(x); . . . ; xk01 p(x)g forms a basis for the code. =): Suppose I is free of rank k with a basis fc 1 c 2 ; . . . ; c g. We know that as an ideal I = hp(x)i where p(x) = f (x)h(x) + 2f (x) for some monic polynomials f (x); h(x); g (x) which are pairwise relatively prime with xn 0 1 = f (x)g (x)h(x). Let s = n 0 deg (p(x)). We first show that k = s. Consider the image I of I under reduction mod2. We claim that jI j = 2k . Suppose that there is a relation a1 c1 + a2 c2 + 1 1 1 + ak c = 0 over 2 , where c denotes the reduction of the vector c modulo 2, which is obtained by reducing each component modulo 2. Viewing the same relation over 4 we get a1 c c1 + a2 c 2 + 1 1 1 + ak c = 2c for some 4 -vector c . Multiplying the last equation by 2, and using the fact that fc1 ; c 2 ; . . . ; c g is linearly independent over 4 , we conclude that 2a1 = 2a2 = 1 1 1 = 2ak = 0. Therefore, a1 ; a2 ; . . . ; ak 2 2 4 , hence a1 ; a2 ; . . . ; ak are all 0 in 2 . This proves that fc 1 ; c 2 ; . . . ; c g forms a basis for I over 2, hence jI j = 2k . On the other hand, the image p(x) of p(x) mod 2 divides xn 0 1, which is a generator of I . This implies that xs p(x), and hence any I
4
...;
x
m
g
p(x)
if
0 deg(
p(x))
=
s
=
k:
Therefore, fp(x); xp(x); . . . ; xk01 p(x)g forms a basis for I . Next we are going to show that h(x) = 1, which will imply that n k p(x)j(x 0 1). Consider the vector x p(x). This can be written as
where f (x)g (x)h(x) = xn 0 1 and jI j = 4deg g(x) 2deg h(x) . Theorem 3 [14], [18]: If n is odd, every ideal I of R is principal. More precisely, I = hp(x)i where p(x) = f (x)h(x) + 2f (x) (or f (x)h(x) + 2f (x)g (x)) and f (x); g (x); h(x) are as above.
s
s01
f
+ 3x + 3x + 3x + 3x + 2x + x + 3
and h(x) = x + 3. Moreover, x23 0 1 = f (x)g (x)h(x) in
, can be written as a linear combination of p(x)g. The latter set is also linearly independent, therefore, it is a basis. This shows that jI j = 2s . Hence, k = s. Since p(x) is the generator of I , any c(x) 2 I can be written as c(x) = t(x)p(x) for some t(x) 2 4 [x]. It follows that the set fp(x); xp(x); x2 p(x); . . .g spans I . Moreover, the set
f
k01
k
x p(x)
for some scalars ai n (x 0 1) where
= i=0
2
i
ai x p(x)
4 . This is equivalent to p(x)t(x) = 0 mod
k t(x)
i
= i=0
with n (x
ak
0 1)j
=
1,
ai x
is a monic polynomial of degree k . This means , i.e., f (x)g (x)h(x)jf (x)(2 + h(x))t(x). Since
p(x)t(x)
deg(p(x)) = deg(f (x)) + deg(h(x))
and n
= deg(f (x)) + deg(h(x)) + deg(g (x))
n 0 deg(p(x)) = k . We see that the degrees of both sides of the last divisibility relation are equal. (Note also that deg(h(x)) = n deg(h(x) + 2)). Therefore, x 0 1 = 6f (x)(h(x) + 2)t(x). Now, by the uniqueness of the factorization of xn 0 1 in 4 [x], we conclude that h(x) = 6(h(x) + 2). Hence h(x) = 1.
deg(g (x)) =
Cyclic codes over 4 corresponding to free modules share many of the properties with their counterparts over fields. The following proposition is an example of this which can be proven easily. Proposition II.1: Let C be a cyclic code of odd length m over 4 with a generator p(x) dividing xm 0 1. Then hp(x)i = hp(x)s(x)i if and only if s(x); xp(x0)1 = 1. Note that when the generator polynomial p(x) of a 4 cyclic code of length n divides xn 0 1, then a generator matrix for the code can be taken to be the matrix with the rows p(x); xp(x);
...;
x
n0deg p(x)
p(x):
If, on the other hand, p(x) is of the form p(x)
=
f (x)h(x)
+ 2f (x)g (x)
(with deg h(x) > 0), then a generator matrix for the code is the matrix with the rows f (x)h(x); xf (x)g (x); .
..;x
k
f (x)g (x);
2f (x)g (x); 2xf (x)g (x); . . . ; 2x
where k1
=
n
k2
=
n
and
0 (deg
f (x)
+ deg h(x) + 1)
0 (deg
f (x)
+ deg g (x) + 1):
k
f (x)g (x)
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
III. QUASI-CYCLIC CODES OVER
2067
4
Quasi-cyclic QC and quasi-twisted (QT) codes over fields have been considered by many authors and a large number of new codes of these types have been discovered. We refer the reader to [1], [4]–[7], [9], [10], [12] and [16] for properties of QC codes and the new codes which are QC. In this correspondence, we consider QC codes over the ring 4 , prove some algebraic results, and obtain some new codes. Definition III.1: Let R be a ring. A linear code of length n = ml over R, i.e., an R-submodule of Rn is called an l-QC code over R if it is invariant under cyclic shifts by l positions. An l-QC code of length n = ml over a ring R is an hxR[x0]1i submodule of ( hxR[x0]1i )l . An r -generator QC code is one with a set of r generators. The main facts about 1-generator QC codes over finite fields as given in [12], [15], and [16]. In this work, we only consider 1-generator QC codes over 4 .
Theorem III.1: Let C be a 1-generator l-QC code of length n = ml, with m odd, over 4 with a generator of the form
( ) = ( ( ) 1( )
g x
g x f
( ) ( ) ...
x ; f2 x g x ;
; f
l (x)g(x))
where
[x] example, negacyclic codes are ideals in the ring hx +1 i . Factorization n n of x +1 in 4 [x] is analogous to that of x 0 1 and is, in fact, obtained from it as follows. Let n be an odd integer, and let xn 0 1 = (x 0 1)a(x)b(x) (which is obtained from the factorization of xn 0 1 in 2 [x] by lifting the factors). Applying the transformation (a ring homomorphism) x 70! 0x to this equation, we get xn + 1 = (x + 1)a(0x)b(0x).
Example IV.1: In Example II.1, we obtained the factorization of 0 1 as
23
x
23
x
01=( ( +2 11
x
+ 3 10 + 2 7 + 6 + 5 + 4 + 2 + 2 + 3) 10 + 3 9 + 3 7 + 3 6 + 3 5 + 2 4 + + 3)( + 3)
11
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
:
Then applying the transformation above, we obtain 23
x
+ 1 = ( 11 + 10 + 2 7 + 3 6 + 5 + 3 4 + 3 2 + 2 + 1) ( 11 + 2 10 + 3 9 + 3 7 + 6 + 3 5 + 2 4 + + 1)( + 1) x
x
x
x
x
x
Any ideal in
hx
x
x
x
x
x
x
x
x
x
x
x
x
:
[ ] +1
i
is generated by an element of the form n x + 1 = f (x)g (x)h(x) in 4 [x]. Theorem II.1 can be easily modified to state that a negacyclic code of length n is a free module of rank k if and only if its generator polynomial divides xn + 1 and has degree n 0 k . An l-QT code is a linear code which is invariant under negacyclic shifts by l positions. More information about QT codes can be found in [1]. We can immediately obtain a result similar to Theorem III.1.
( ) = ( ) ( ) + 2 ( ) where
p x
f x h x
f x
( )j( m 0 1) ( ) i ( ) 2 4 [ ] h m 0 1i and ( i ( ) ( )) = 1, ( ) = xg(x0)1 for all 1 . Then C is a free 4 -module of rank 0 deg ( ) and 1 L (C ), where Theorem IV.1: Let C be a 1-generator -QT code of length = , and L (C ) are the minimum Lee weights of the cyclic code generated with odd, over 4 with a generator of the form by ( ) and the code C , respectively. ( ) = ( ( ) 1 ( ) 2 ( ) ( ) . . . l ( ) ( )) Proof: Let 1 . For fixed consider the following th projection map on an -QC code C of length = : m + 1i, and where ( )j( m + 1) ( ) i ( ) 2 4[ ] h 5i : n4 ! m4 ( i ( ) ( )) = 1, ( ) = xg(x+1) for all 1 . Then C is a free 0 deg ( ) and 1 L (C ), where and 4 -module of rank ( 0 1 . . . ml01 ) ! (i01)m 1+(i01)m . . . m01+(i01)m ( C ) are the minimum Lee weights of the negacyclic code generated L Observe that 5i (C ) is a cyclic code generated by h i ( ) ( )i for by ( ) and the code C , respectively. all 1 . We have that one of the components becomes zero if and only if all the others do because ( ) i ( ) ( ) = 0 if and only if ( )j( ( ) i ( )) (if ( ) 6= 0), which implies that ( )j ( ) since V. NEW CODES AND THEIR GENERATORS ( i ( ) ( )) = 1. So, ( ) j ( ) ( ) = 0 for all . Therefore, if is Theorem 5 was the basis of the search method of [16] and [1] where a nonzero codeword in C then 5i ( ) 6= 0 for all . Since h i ( ) ( )i = they many new QC and QT codes. The analogous Theorems III.1 h ( )i, 5i (C ) is a cyclic code with generator polynomial ( ), and (and found IV.1) for 4 codes is the basis of our method of search. We start every nonzero codeword has Lee weight (by assumption). Hence, m 0 1 in 2 [ ] for an odd integer . Then a nonzero codeword in C has a weight greater than or equal to 1 . with a polynomial 2 ( )j m 0 1 in 4 [ ] (or dividing m +1). Moreover, it can be shown, similar to the cyclic code case, that elements we compute its Hensel lift ( )j ( ) ( ) . . . n0(deg(g(x)01) ( ) form a basis for the code. In Note that in this case the cyclic code generated by ( ) is a free module of rank 0 deg ( ). Then we search over 1-generator 4 -QC (or QT) fact, if a relation codes of the form ( ( ) 1 ( ) ( ) 2 ( ) . . . ( ) l ( )) of length deg g (x)01 = . In fact, we always work with the case = 2 in this search. i ( )=0 with i 2 4 i Finally, we map our codes to 2n via the Gray map and get binary g x
f
x
;
x ; h x
g x ; f
x
x = x
h x
i
m
g x
l
d
l
d
d
l
d
g x
i
l
i
l
g x
i
n
ml
g x
f
c ; c ;
; c
i
c
; c
;
f
; c
p x f
x
ml
p x f
x
f
x g x
h x
p x
x ;f
x g x ;
; g x ; f
x ; h x
;f
x g x
x = x
x
h x
i
m
g x
l
d
l
d
d
d
g x
x g x
p x
x ; h x
g x f
:
l
p x f
h x
n
m
x g x
c
j
c
i
f
x g x g x
g x
d
l
g
d
x
x
x
g x
g x ; g x ; xg
g x
g x f
a x g x
;
n
a
g x 01
i
i
( )=0
a x g x
i=0 i bi x g (x) 6= 0, then neither is i
i i bi x g (x)=0.
IV. NEGACYCLIC AND QUASI-TWISTED CODES A 4 -linear code is called negacyclic if it is invariant under the negacyclic shift
(0
c ; c1 ;
...
; c
n01 ) 70! (0cn01 ; c0 ;
...
x ; g x f
x ;
; g x f
ml
x
l
F2
exists (with m-dimensional vectors), then a similar relation deg ( )
x
g x
n
holds in n 4 . Also, if
x
g g x
; x
i=0
m
x
; c
n02 ):
Negacyclic codes over 4 have recently been investigated in [19]. Most of the results about cyclic codes are also true for negacyclic code. For
codes. The parameters of new codes, their generators, and Lee weight enumerators follow. In some cases, we have been able to determine that resulting binary codes are not linear. For each of the codes 1–3, we considered linear span of k vectors (for appropriate k ) and we found words of weights not appearing in the weight enumerators of these codes, showing that these codes are not linear. In general, however, determining linearity of Gray image of a 4 code is not trivial. 1) A (92;
224 28) binary code: Let ( )= 11 +3 10 +2 7 + 6 + 5 + 4 + 2 +2 +3j 23 0 1
g x
;
x
x
x
x
x
x
x
x
x
considered in the Example I.1. Note that this is the lift of the famous binary Golay code to 4 which is studied in details in [8]. And let
( )=2 2 +2 3 + 4 +2 5 +2 7 +2 8 +2 9 +3 10 +
f x
x
x
x
x
x
x
x
x
11
x
:
2068
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
Then the 4 -QC code with the generators (g (x); g (x)f (x)) has length 46, rank 12, and minimum Lee weight 28 (with computer calculations). In fact, its Lee weight enumerator is x
92
64 28
+ 3082x
62 30
+ 10120x
y
58 34
+ 121072x
y
56 36
+ 320068x
52 40
+ 1294164x
y
+ 1969168x + 2538878x + 690368x
y y
34 58
+ 121072x
y
28 64 y
44 48 y
38 54
+ 38111x
+y
54 38
y
y
32 60
92
y
y
+ 690368x
50 42
40 52
+ 1294164x + 3082x
y
46 46
+ 2807152x
60 32
+ 38111x
y
y
48 44
+ 2538878x
y
42 50
+ 1969168x
y
36 56
+ 320068x
y
g2 (x)
6
x
x 01 h (x) ,
=
j
5
+x +1
63
x
+ 10120x
252
132 120
+ 2520x
y
0 1 mod 2
124 128
+ 63x
252
136 116
+1128x
y
128 124
y
+63x
y
126 126
+64x
118 134
+3040x
134 118
+3040x
y
y
y
116 136
+ 1512x
y
:
132 120
+1880x
124 128
+63x
116 132
+1128x
y
+y
y
252
y
120 132
+1880x
y
:
4) A (60; 28 ; 28) binary code: Let h2 (x) = x4 + x + 1jx15 0 1 1 and g2 (x) = xh (0 x) in 2 [x]. Let g (x) be the Hensel lift of g2 (x). Then the 4 -QC code of length 30 generated by 2 3 (g (x); g (x)f (x)) where f (x) = 2x + x has rank 4 and minimum Lee weight 28. Therefore, its Gray image is a 8 (60; 2 ; 28)-code. This code is necessarily nonlinear because its minimum distance exceeds the largest possible minimum distance of a linear code with these parameters [2]. Moreover, it gives the parameters of the best known binary nonlinear code [13]. The Lee weight enumerator of this code is 60
x
32 28
+ 180x
y
28 32
+ 15x
y
32 28
+ 180x
y
24 36
+ 60x
y
30 30
+ 16x
y
y
y
26 34
+ 120x
y
24 36
+ 60x
y
:
6) A (62; 29 ; 28) binary code: This is obtained by extending the previous code by a parity check and has a larger minimum distance than the best known binary linear code with parameters [62; 9; 27]. 7) A (84; 213 ; 34) binary code: Let g (x)
6
5
4
2
3
2
= (x + 3x + 3x + 3x + 2x + 1)(x + 2x + x + 1) 3
2
2
(x + x + 2x + 1)(x + 3x + 1) 21
which divides x
6
f (x)
=
x
h(x)
=
x
+1
in
4 [x],
and let
5
4
2
+ 2x + 3x + 3x + 3x + 1
+1
21
3) A (252; 214 ; 116) binary code: With the same setting and x) the notation as in the last example, let g 0 (x) = gx(+3 . And let 0 4 5 6 f (x) = x + x + 3x . Then the 4 -QC code generated by 0 0 0 (g (x); g (x)f (x)) has length 126, rank 7, and minimum Lee weight 116. Its Gray image, a (252; 214 ; 116) nonlinear code, looks much better than the comparable [252; 14; 112] binary code. The Lee weight enumerator of this code is x
34 26
+ 120x
y
and let g (x) be the Hensel lift of g2 (x) in 4 [x]. Then the 4 -QC code of length 126 generated by 3 4 (g (x); g (x)f (x)) where f (x) = x + 2x has rank 6 and minimum Lee weight 120. Its Gray image is a (252; 212 ; 120) nonlinear code, while the best known binary linear code of length 252 and dimension 12 has minimum distance 118. The upper bound for a binary linear code of this length and dimension is 120 [2]. However, this upper bound does not prove the existence of a code achieving it. The Lee weight enumerator of this code is x
60
x
+15x
Therefore, its Gray image is a (92; 2 ; 28) binary code, which is nonlinear. This code is not only better than the comparable (best known) binary linear code [92; 24; 26] [2], but also establishes a new record among all nonlinear binary codes [13]. In terms of the notation of [13], this improves A(92; 28) = 23 A(91; 27) from 2 to 224 where A(n; d) denotes the size of the largest binary code of length n and minimum distance d. In other words, this code contains twice as many codewords as the previously best known comparable code.
=
1
30 62
24
h2 (x)
C
28 32
:
2) A (252; 212 ; 120) binary code: Let
3 2 = (3x + x )g (x)h(x) + 2g (x)f (x), respectively. Then is free of rank 4 but C2 is not; it is of type 44 21 . Then the 4 -QC code generated by (p1 (x); p2 (x)) has length 30, size 9 2 , and minimum Lee weight 26. Its Gray image gives a binary (60; 29 ; 26)-code, which has the parameters of the best known binary linear code which is also optimal. This code has fewer codewords than the comparable best known nonlinear binary code [13]. The Lee weight enumerator is p2 (x)
:
5) A (60; 29 ; 26) binary code: Let g (x) be as in the last example and let h(x) = x + 3, and f (x) = gx(x)h0(1x) . Consider the 4 cyclic codes C1 and C2 generated by p1 (x) = 3x3 g (x)h(x) and
then x + 1 = f (x)g (x)h(x) and this factorization can be obtained from that of x21 0 1. The polynomial g (x) generates a negacyclic code of length 21 of rank 7. Let g (x )
= (g (x)f1 (x);
g (x)f2 (x))
where 6
5
f1 (x)
= 3x + x
f2 (x)
=
and 6
5
4
3
2
+ 2x + x + x + 2x + 1:
x
The QT code whose generator matrix has the rows g (x ); g (x ); xg
...;
6
x g (x );
2x
7
g (x )
13
has length 42, size 2 , and minimum Lee weight 34. Therefore, its Gray image gives a binary (84; 213 ; 34)-code which has a larger minimum distance than the best known [84; 13; 33] binary code. The Lee weight enumerator of this code is 84
x
50 34
+ 504x
y
42 42
+ 1008x
y
34 50
+ 504x
y
48 36
+ 609x
y
40 44
+ 798x
+y
84
y
46 38
+ 1680x
y
38 46
+ 1680x
y
44 40
+ 798x
y
36 48
+ 609x
y
:
8) A (252; 213 ; 118) binary code: Let f (x) = x6 + x5 + 2x3 + 1, 63 h(x) = x + 1 which both divide x + 1 in 4 [x], and let x +1 g (x) = f (x)h(x) . Let g (x ) = (g (x)f1 (x); g (x)f2 (x)) where 6 5 5 4 f1 (x) = x + x and f2 (x) = x + 3x . Then the 4 QT code with generator matrix having the rows g (x ); g (x) ; xg
...;
6
x g g(x );
2x
7
g (x )
13
has length 126, size 2 , and minimum Lee weight 118. Therefore, its Gray image is a (252; 213 ; 118) binary code with a larger minimum distance than the best known [252; 13; 116] binary linear code. The Lee weight enumerator is as follows: 252
x
134 118
+ 2016x
y
132 120
+ 2520x
124 128
+63x
y
y
126 126
+ 64x
118 134
+ 2016x
y
y
116 136
+ 1512x
y
:
9) A (254; 213 ; 120) binary code: This code is obtained by extending the previous code by a parity check and then taking the Gray image. It has a better minimum distance than the best known [254; 13; 118] binary linear code and has the parameters of an optimal linear code for this length and the size (which is currently hypothetical and yet to be constructed).
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
ACKNOWLEDGMENT The authors wish to thank Prof. J. Ferrar of the Department of Mathematics, The Ohio State University, for his useful discussions and the referees for the suggestions. REFERENCES [1] N. Aydin, I. Siap, and D. K. Ray-Chaudhuri, “The structure of 1-generator quasitwisted codes and new linear codes,” Des., Codes Cryptogr., vol. 23, no. 3, pp. 313–326, December 2001. [2] A. E. Brouwer. Linear code bounds. [Online]. Available: http://www. win.tue.nl/aeb/voorlincod.html [3] A. R. Calderbank and G. McGuire, “Construction of a (64; 2 ; 12) code via Galois rings,” Des. Codes Cryptogr., vol. 10, no. 2, pp. 157–165, 1997. [4] Z. Chen, “Six new binary quasicyclic codes,” IEEE Trans. Inform. Theory, vol. 40, pp. 1666–1667, Sept. 1994. [5] R. N. Daskalov, T. A. Gulliver, and E. Metodieva, “New good quasicyclic ternary and quaternary linear codes,” IEEE Trans. Inform. Theory, vol. 43, pp. 1647–1650, Sept. 1997. , “New ternary linear codes,” IEEE Trans. Inform. Theory, vol. 45, [6] pp. 1687–1688, July 1999. [7] P. P. Greenough and R. Hill, “Optimal ternary quasicyclic codes,” Des. Codes, Cryptogr., vol. 2, pp. 81–91, 1992. [8] M. Greferath and E. Viterbo, “On - and -linear lifts of the Golay code,” IEEE Trans. Inform. Theory, vol. 45, pp. 2524–2527, Nov. 1999. [9] T. A. Gulliver and V. K. Bhargava, “Nine good rate (m 1)=pm quasicyclic codes,” IEEE Trans. Inform. Theory, vol. 38, pp. 1366–1369, July 1992. [10] , “New good rate (m 1)=pm ternary and quaternary quasicyclic codes,” Des., Codes Cryptogr., vol. 7, pp. 223–233, 1996. [11] A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z -linearity of Kerdock, Preperata, Goethals, and related codes,” IEEE Trans. Inform. Theory, vol. 40, pp. 301–319, Mar. 1994. [12] K. Lally and P. Fitzpatrick, “Construction and classification of quasicyclic codes,” in Proc. Workshop on Coding and Cryptography (WCC 99), Paris, France, Jan. 11–14, 1999. [13] S. Litsyn. Table of nonlinear binary codes. [Online]. Available: http://www.eng.tau.ac.il/litysn/tableand/index.html [14] V. S. Pless and Z. Qian, “Cyclic codes and quadratic residue codes over ,” IEEE Trans. Inform. Theory, vol. 42, pp. 1594–1600, Sept. 1996. [15] G. E. Séguin and G. Drolet, “The theory of 1-generator quasicyclic codes,” preprint, 1990. [16] I. Siap, N. Aydin, and D. K. Ray-Chaudhuri, “New ternary quasicyclic codes with better minimum distances,” IEEE Trans. Inform. Theory, vol. 46, pp. 1554–1558, July 2000. [17] J. H. van Lint, Introduction to Coding Theory. Berlin, Germany: Springer-Verlag, 1999. [18] Z. X. Wan, Quaternary Codes. Singapore: World Scientific, 1997. [19] J. Wolfmann, “Negacyclic and cyclic codes over ,” IEEE Trans. Inform. Theory, vol. 45, pp. 2527–2532, Nov. 1999. [20] M. Yamada, “Distance-regular graphs of girth 4 over an extension ring of Z/4Z,” Graphs Comb., vol. 6, pp. 381–394, 1990.
0
0
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The -ary Image of Some -ary Cyclic Codes: Permutation Group and Soft-Decision Decoding Jérôme Lacan and Emmanuelle Delpeyroux Abstract—Using a particular construction of generator matrices of the -ary image of -ary cyclic codes, it is proved that some of these codes are invariant under the action of particular permutation groups. The equivalence of such codes with some two-dimensional (2-D) Abelian codes and cyclic codes is deduced from this property. These permutations are also used in the area of the soft-decision decoding of some expanded Reed–Solomon (RS) codes to improve the performance of generalized minimum-distance decoding. Index Terms—Permutation groups, -ary image of soft-decision decoding.
-ary cyclic codes,
I. INTRODUCTION One important area in coding theory is concerned with codes which are invariant under a set of permutations. For example, any linear code of length n over q (i.e., vector subspace of ( q )n ) which has the property of being invariant under the cyclic permutation (shift) is called a cyclic code. This property allows us to consider such a code as a principal ideal in the algebra n q [z ]=(z 0 1). Let (i; j ) denote the greatest common divisor of two integers i and j . When (n; q ) = 1, a cyclic code is entirely defined by the set of its nonzeros, i.e., the nth roots of unity such that at least one codeword polynomial c(x) does not evaluate to zero in this point. The parameters of such a code are usually denoted by [n; k], where k is the dimension of the code considered as a q -vector subspace. Note that k is also the number of nonzeros. In order to distinguish different codes of the same dimension, we denote a code by [n; N Z ]q , where N Z is the set of nonzeros. A generator polynomial g (z ) of such a code is a polynomial codeword such that each codeword c(z ) can be expressed as a product c(z ) = g (z ) 2 u(z ) where u(z ) is a polynomial of degree less than k . From such a generator polynomial, we can obtain a generator matrix G of this code defined as follows:
1 1 1 k01 g(z) where GT denotes the transpose of G. G
T =
g (z ) zg (z )
z
An Abelian code over q is a vector subspace of ( q )mn invariant under the shift of the rows and the columns where we consider each codeword as an m 2 n-matrix [6]. Many researchers have worked on codes which are invariant under particular groups of permutations such as, for example, the general linear group [1]–[3] or the Mathieu group [5, Ch. 20]. The first problem we address in this correspondence is that of determining a group of permutations under whose action the q -ary image of some q m -ary cyclic codes is invariant. This property is used to obtain new results on these codes such as the invariance under particular permutations or the equivalence with cyclic and two-dimensional (2-D) Abelian codes. Permutations that fix a code have an interesting application in the area of soft-decision decoding of the q -ary image (the so-called expanded image) of the Reed–Solomon (RS) codes. For these codes,
Manuscript received September 5, 2000; revised September 7, 2001. J. Lacan is with the Département de Mathématiques Appliquées et d’Informatique, ENSICA, 31056 Toulouse, France (e-mail: [email protected]). E. Delpeyroux is with the Département d’Informatique, ICAM, 31300 Toulouse, France (e-mail: [email protected]). Communicated by R. Koetter, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(02)05155-6. 0018-9448/02$17.00 © 2002 IEEE
