Double Circulant Self-dual Codes Research Articles

Generalized Cyclotomic Double Circulant Self-Dual Codes of Length 4p

Constructions of some double circulant self-dual codes on the residue ring $Z_{2p}$ , where $p$ is a prime and satisfies $p \equiv 3 \bmod {4}$ , are proposed. Two new structures of these codes are formed from the theory of generalized cyclotomy. Moreover, both [12, 6, 4] and [44, 22, 8] optimal self-dual codes over GF(2) and a family of [28, 14, 9] highest known self-dual codes over GF(4) are given.

On extremal double circulant self-dual codes of lengths 90–96

A classification of extremal double circulant self-dual codes of lengths up to 88 is known. We extend this classification to length 96. We give a classification of extremal double circulant self-dual codes of lengths 90, 92, 94 and 96. We also classify double circulant self-dual codes with parameters [90, 45, 14] and [96, 48, 16]. In addition, we demonstrate that no double circulant self-dual [90, 45, 14] code has an extremal self-dual neighbor, and no double circulant self-dual [96, 48, 16] code has a self-dual neighbor with minimum weight at least 18.

Fourth Power Residue Double Circulant Self-Dual Codes

Quadratic residue codes are a well-known class of codes. In this paper, we consider the constructions of self-dual codes by higher power residues, especially fourth power residues. New infinite families of self-dual codes over GF(2), GF(3), GF(4), GF(8), and GF(9) are introduced. Some of them have better minimum weight than previously known codes. We also give general results related to the automorphism group of some of these codes.

Mutually disjoint t-designs and t-SEEDs from extremal doubly-even self-dual codes

It is known that extremal doubly-even self-dual codes of length $$n\equiv 8$$ n ? 8 or $$0\ (\mathrm {mod}\ 24)$$ 0 ( mod 24 ) yield 3- or 5-designs respectively. In this paper, by using the generator matrices of bordered double circulant doubly-even self-dual codes, we give 3-(n, k; m)-SEEDs with (n, k, m) $$\in \{(32,8,5), (56,12,9), (56,16,9), (56,24,9), (80,16,52)\}$$ ? { ( 32 , 8 , 5 ) , ( 56 , 12 , 9 ) , ( 56 , 16 , 9 ) , ( 56 , 24 , 9 ) , ( 80 , 16 , 52 ) } . With the aid of computer, we obtain 22 generator matrices of bordered double circulant doubly-even self-dual codes of length 48, which enable us to get 506 mutually disjoint 5-(48, k, $$\lambda $$ ? ) designs for (k, $$\lambda $$ ? )=(12, 8),(16, 1356),(20, 36176). Moreover, this implies 5-(48, k; 506)-SEEDs for $$k=12, 16, 20, 24$$ k = 12 , 16 , 20 , 24 .

Construction of new self-dual codes over $GF(5)$ using skew-Hadamard matrices

In this paper, we give optimal self-dual codes over $GF(5)$ for lengths $24$, $40$, $48$ and $56$. In particular, new inequivalent $[48, 24]$ and $[56, 28]$ self-dual codes over $GF(5)$ whose minimum weights are $14$ and $16$, are constructed using skew-Hadamard matrices of order $24$ and $28$, thus improving the only known quadratic double circulant self-dual codes of length $48$ and $56$. Moreover, $[80, 40]$ and $[88, 44]$ self-dual codes whose minimum weights are $17$ and $19$ over $GF(5)$, are constructed for the first time. These codes are derived from skew-Hadamard matrices of order $40$ and $44$, respectively. Finally, a new $[56, 28, 17]$ self-dual code is constructed over $GF(7)$ having the highest minimum weight among $[56, 28]$ self-dual codes. This new optimal code is constructed from a skew-Hadamard-matrix of order $28$, for the first time.

Classification of extremal double circulant self-dual codes of lengths 74–88

A classification of all extremal double circulant self-dual codes of lengths up to 72 is known. In this paper, we give a classification of all extremal double circulant self-dual codes of lengths 74–88.

Optimal double circulant self-dual codes over F/sub 4/

Optimal double circulant self-dual codes over F/sub 4/ have been found for each length n/spl les/40. For lengths n/spl les/14, 20, 22, 24, 28, and 30, these codes are optimal self-dual codes. For length 26, the code attains the highest known minimum weight. For n/spl ges/32, the codes presented provide the highest known minimum weights. The [36,18,12] self-dual code improves the lower bound on the highest minimum weight for a [36,18] linear code.

New Optimal Self-Dual Codes over GF(7)

In this paper, double circulant self-dual codes over GF(7) are presented, including [12,6,6] codes which are new optimal codes. It is shown that the supports of the codewords of weights 9, 10 and 11 in double circulant [20,10,9] codes form 3-designs. For larger lengths, some good self-dual codes are constructed from weighing matrices.

Double circulant self-dual codes over Z/sub 2k/

Recently there has been tremendous interest in self-dual codes over finite rings, specifically the rings Z/sub 2k/. In this paper, we investigate double circulant self-dual codes over Z/sub 2k/ for small k and short lengths. In particular, we give a classification of the extremal double circulant codes. Using invariant theory, a basis for the space of invariants to which the Hamming weight enumerators belong is given for self-dual codes over Z/sub 6/, Z/sub 8/, Z/sub 10/, and Z/sub 12/.

Classification of Extremal Double Circulant Self-Dual Codesof Lengths 64 to 72

Recently extremal double circulant self-dual codes have been classified for lengths n ≤ 62. In this paper, a complete classification of extremal double circulant self-dual codes of lengths 64 to 72 is presented. Almost all of the extremal double circulant singly-even codes given have weight enumerators for which extremal codes were not previously known to exist.

Weight Enumerators of Double Circulant Codes and New ExtremalSelf-Dual Codes

In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44,22,8] double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.