Weight Enumerators of Double Circulant Codes and New Extremal Self-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.     

Classification of Extremal Double Circulant Formally Self-Dual Even Codes

Formally self-dual even codes have recently been studied. Double circulant even codes are a family of such codes and almost all known extremal formally self-dual even codes are of this form. In this paper, we classify all extremal double circulant formally self-dual even codes which are not self-dual. We also investigate the existence of near-extremal formally self-dual even codes.     

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.     

Near-Extremal Formally Self-Dual Even Codes of Lengths 24 and 32

The weight enumerator of a formally self-dual even code is obtained by the Gleason theorem. Recently, Kim and Pless gave some restrictions on the possible weight enumerators of near-extremal formally self-dual even codes of length divisible by eight. In this paper, the weight enumerators for which there is a near-extremal formally self-dual even code are completely determined for lengths 24 and 32, by constructing new near-extremal formally self-dual codes. We also give a classification of near- extremal double circulant codes of lengths 24 and 32. Communicated by: P. Fitzpatrick     

2n Bordered constructions of self-dual codes from group rings

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual codes by combining many of the previously used techniques. The purpose of this is to construct self-dual codes that were missed using classical construction techniques by constructing self-dual codes with different automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct some extremal self-dual codes of length 64 and 68, constructing 30 new extremal self-dual codes of length 68.     

New Extremal Type I Codes of Lengths 40, 42, and 44

It is known that it is possible to construct a generator matrix for a self-dual code of length 2n+2 from a generator matrix of a self-dual code of length 2n. With the aid of a computer, we construct new extremal Type I codes of lengths 40, 42, and 44 from extremal self-dual codes of lengths 38, 40, and 42 respectively. Among them are seven extremal Type I codes of length 44 whose weight enumerator is 1+224y ⁸+872y ¹⁰+·. A Type I code of length 44 with this weight enumerator was not known to exist previously.     

Construction of Extremal Type II Codes over ℤ4

In this paper, we give a pseudo-random method to construct extremal Type II codes overℤ₄ . As an application, we give a number of new extremal Type II codes of lengths 24, 32 and 40, constructed from some extremal doubly-even self-dual binary codes. The extremal Type II codes of length 24 have the property that the supports of the codewords of Hamming weight 10 form 5−(24,10,36) designs. It is also shown that every extremal doubly-even self-dual binary code of length 32 can be considered as the residual code of an extremal Type II code over ℤ₄.     

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. Received: June 24, 1996 / Revised: February 28, 1997     

Self-dual 2-quasi-cyclic codes and dihedral codes

We characterize the structure of 2-quasi-cyclic codes over a finite field F by the so-called Goursat Lemma. With the characterization, we exhibit a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code. And we obtain a necessary and sufficient condition for a self-dual 2-quasi-cyclic code being a dihedral code (if $\mathrm{char}F=2$), or a consta-dihedral code (if $\mathrm{char}F\ne 2$). As a consequence, any self-dual 2-quasi-cyclic code generated by one element must be (consta-)dihedral. In particular, any self-dual double circulant code must be (consta-)dihedral. We also obtain necessary and sufficient conditions under which the three classes (the self-dual double circulant codes, the self-dual 2-quasi-cyclic codes, and the self-dual (consta-)dihedral codes) of codes coincide with each other.     

Computational results of duadic double circulant codes

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over $\mathbb{F}_{2}$ , $\mathbb{F}_{3}$ , $\mathbb{F}_{4}$ , $\mathbb{F}_{5}$ , and $\mathbb{F}_{7}$ . In particular, we find a new ternary self-dual [76,38,18] code and easily rediscover optimal binary self-dual codes with parameters [66,33,12], [68,34,12], [86,43,16], and [88,44,16] as well as a formally self-dual binary [82,41,14] code.     

The nonexistence of near-extremal formally self-dual codes

A code is called formally self-dual if and have the same weight enumerators. There are four types of nontrivial divisible formally self-dual codes over , and . These codes are called extremal if their minimum distances achieve the Mallows-Sloane bound. S. Zhang gave possible lengths for which extremal self-dual codes do not exist. In this paper, we define near-extremal formally self-dual (f.s.d.) codes. With Zhang’s systematic approach, we determine possible lengths for which the four types of near-extremal formally self-dual codes as well as the two types of near-extremal formally self-dual additive codes cannot exist. In particular, our result on the nonexistence of near-extremal binary f.s.d. even codes of any even length n completes all the cases since only the case 8|n was dealt with by Han and Lee.      

Extremal Self-Dual Codes over \mathbb{F} 2 × \mathbb{F} 2

In this paper, it is shown that extremal (Hermitian) self-dual codes over ₂ × ₂ exist only for lengths 1, 2, 3, 4, 5, 8 and 10. All extremal self-dual codes over ₂ × ₂ are found. In particular, it is shown that there is a unique extremal self-dual code up to equivalence for lengths 8 and 10. Optimal self-dual codes are also investigated. A classification is given for binary [12, 7, 4] codes with dual distance 4, binary [13, 7, 4] codes with dual distance 4 and binary [13, 8, 4] codes with dual distance 4.     

New Extremal Type II Codes Over ℤ4

Recently Type II codes over ℤ₄ have been introduced as self-dual codes containing the all-one vector with the property that all Euclidean weights are divisible by eight. The notion of extremality for the Euclidean weight has been also given. In this paper, we give two methods for constructing Type II codes over ℤ₄. By these methods, new extremal Type II codes of lengths 16, 24, 32 and 40 are constructed from weighing matrices.     

On the Classification of Extremal Even Formally Self-Dual Codes

Bachoc bachoc has recently introduced harmonic polynomials for binary codes. Computing these for extremal even formally self-dual codes of length 12, she found intersection numbers for such codes and showed that there are exactly three inequivalent [12,6,4] even formally self-dual codes, exactly one of which is self-dual. We prove a new theorem which gives a generator matrix for formally self-dual codes. Using the Bachoc polynomials we can obtain the intersection numbers for extremal even formally self-dual codes of length 14. These same numbers can also be obtained from the generator matrix. We show that there are precisely ten inequivalent [14,7,4] even formally self-dual codes, only one of which is self-dual.     

A complete classification of ternary self-dual codes of length 24

Ternary self-dual codes have been classified for lengths up to 20. At length 24, a classification of only extremal self-dual codes is known. In this paper, we give a complete classification of ternary self-dual codes of length 24 using the classification of 24-dimensional odd unimodular lattices.     

Extremal Ternary Self-Dual Codes Constructed from Negacirculant Matrices

In this paper, a construction of ternary self-dual codes based on negacirculant matrices is given. As an application, we construct new extremal ternary self-dual codes of lengths 32, 40, 44, 52 and 56. Our approach regenerates all the known extremal self-dual codes of lengths 36, 48, 52 and 64. New extremal ternary quasi-twisted self-dual codes are also constructed. Supported by an NSERC discovery grant and a RTI grant. Supported by an NSERC discovery grant and a RTI grant. A summer student Chinook Scholarship is greatly appreciated.     

Binary self-dual codes with automorphisms of order 23

The only example of a binary doubly-even self-dual [120,60,20] code was found in 2005 by Gaborit et al. (IEEE Trans Inform theory 51, 402–407 2005). In this work we present 25 new binary doubly-even self-dual [120,60,20] codes having an automorphism of order 23. Moreover we list 7 self-dual [116,58,18] codes, 30 singly-even self-dual [96,48,16] codes and 20 extremal self-dual [92,46,16] codes. All codes are new and present different weight enumerators.      

On the Covering Radius of Ternary Extremal Self-Dual Codes

In this paper, we investigate the covering radius of ternary extremal self-dual codes. The covering radii of all ternary extremal self-dual codes of lengths up to 20 were previously known. The complete coset weight distributions of the two inequivalent extremal self-dual codes of length 24 are determined. As a consequence, it is shown that every extremal ternary self-dual code of length up to 24 has covering radius which meets the Delsarte bound. The first example of a ternary extremal self-dual code with covering radius which does not meet the Delsarte bound is also found. It is worth mentioning that the found code is of length 32.     

An efficient algorithm for constructing reversible quasi-cyclic codes via Chinese remainder theorem

Regarding quasi-cyclic codes as certain polynomial matrices, we show that all reversible quasi-cyclic codes are decomposed into reversible linear codes of shorter lengths corresponding to the coprime divisors of the polynomials with the form of one minus x to the power of m. This decomposition brings us an efficient method to construct reversible quasi-cyclic codes. We also investigate the reversibility and the self-duality of the linear codes corresponding to the coprime divisors of the polynomials. Specializing to the cases where the number of cyclic sections is not more than two, we give necessary and sufficient conditions for the divisors of the polynomials for which the self-dual codes are reversible and the reversible codes of half-length-dimension are self-dual. Our theorems are utilized to search reversible self-dual quasi-cyclic codes with two cyclic sections over binary and quaternary fields of lengths up to seventy and thirty-six, respectively, together with the maximums of their minimum weights.