The Invariants of the Clifford Groups

The automorphism group of the Barnes-Wall lattice L _m in dimension 2 ^m (m ; 3) is a subgroup of index 2 in a certain Clifford group of structure 2 ₊ ^1+2m . O ⁺(2m,2). This group and its complex analogue of structure .Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge@apos;s 1996 result that the space of invariants for of degree 2k is spanned by the complete weight enumerators of the codes , where C ranges over all binary self-dual codes of length 2k; these are a basis if m k - 1. We also give new constructions for L _m and : let M be the -lattice with Gram matrix . Then L _m is the rational part of M ^m, and = Aut(M^m). Also, if C is a binary self-dual code not generated by vectors of weight 2, then is precisely the automorphism group of the complete weight enumerator of . There are analogues of all these results for the complex group , with doubly-even self-dual code instead of self-dual code.     

Singly-Even Self-Dual Codes of Length 40

All singly-even self-dual [40,20,8] binary codes which have an automorphism of prime order are obtained up to equivalence. There are two inequivalent codes with an automorphism of order 7 and 37 inequivalent codes with an automorphism of order 5. These codes have highest possible minimal distance and some of them are the first known codes with weight enumerators prescribed by Conway and Sloane.     

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 binary self-dual codes of lengths 60, 62, 64 and 66 having an automorphism of order 9

A method for constructing binary self-dual codes having an automorphism of order p ² for an odd prime p is presented in (S. Bouyuklieva et al. IEEE. Trans. Inform. Theory, 51, 3678–3686, 2005). Using this method, we investigate the optimal self-dual codes of lengths 60 ≤ n ≤ 66 having an automorphism of order 9 with six 9-cycles, t cycles of length 3 and f fixed points. We classify all self-dual [60,30,12] and [62,31,12] codes possessing such an automorphism, and we construct many doubly-even [64,32,12] and singly-even [66,33,12] codes. Some of the constructed codes of lengths 62 and 66 are with weight enumerators for which the existence of codes was not known until now.      

On circulant self-dual codes over small fields

We construct self-dual codes over small fields with q = 3, 4, 5, 7, 8, 9 of moderate length with long cycles in the automorphism group. With few exceptions, the codes achieve or improve the known lower bounds on the minimum distance of self-dual codes.      

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.     

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.     

The Classification of Some Perfect Codes

Perfect 1-error correcting codes C in Z ₂ ⁿ , where n=2 ^m–1, are considered. Let ; denote the linear span of the words of C and let the rank of C be the dimension of the vector space . It is shown that if the rank of C is n–m+2 then C is equivalent to a code given by a construction of Phelps. These codes are, in case of rank n–m+2, described by a Hamming code H and a set of MDS-codes D _h , h H, over an alphabet with four symbols. The case of rank n–m+1 is much simpler: Any such code is a Vasil'ev code.     

The Automorphism Groups of BCH Codes and of Some Affine-Invariant Codes Over Extension Fields

Affine-invariant codes are extended cyclic codes of length p ^m invariant under the affine-group acting on . This class of codes includes codes of great interest such as extended narrow-sense BCH codes. In recent papers, we classified the automorphism groups of affine-invariant codes berg, bech1. We derive here new results, especially when the alphabet field is an extension field, by expanding our previous tools. In particular we complete our results on BCH codes, giving the automorphism groups of extended narrow-sense BCH codes defined over any extension field.     

Classification of Formally Self-Dual Even Codes of Lengths up to 16

In this note, we give the complete classification of binary formally self-dual even codes of lengths 10, 12, 14 and 16. There are exactly fourteen, 29, 99 and 914 inequivalent such codes of lengths 10, 12, 14 and 16, respectively. This completes the classification of formally self-dual even codes of lengths up to 16. The first example of formally self-dual even code with a trivial automorphism group is also found. This shows that 16 is the smallest length for which there is a formally self-dual even code with a trivial automorphism group.     

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.     

An Enumeration of Binary Self-Dual Codes of Length 32

A binary self-dual code of length 2k is a (2k, k) binary linear code C with the property that every pair of codewords in C are orthogonal. Two self-dual codes, C ₁ and C ₂, are equivalent if and only if there is a permutation of the coordinates of C ₁ that takes C ₁ into C ₂. The automorphism group of a binary code C is the set of all permutations of the coordinates of C that takes C into itself.The main topic of this paper is the enumeration of inequivalent binary self-dual codes. We have developed algorithms that will take lists of inequivalent small codes and produce lists of larger codes where each inequivalent code occurs only a few times. We have defined a canonical form for codes that allowed us to eliminate the overenumeration. So we have lists of inequivalent binary self-dual codes of length up to 32. The enumeration of the length 32 codes is new. Our algorithm also finds the size of the automorphism group so that we can compute the number of distinct binary self-dual codes for a specific length. This number can also be found by counting and matches our total.     

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.      

The classification of self-dual modular codes

A classification method of self-dual codes over Zm is given. If m=rs with relatively prime integers r and s, then the classification can be accomplished by double coset decompositions of Sn by automorphism groups of self-dual codes over Zr and Zs. We classify self-dual codes of length 4 over Zp for all primes p in terms of their automorphism groups and then apply our method to classify self-dual codes over Zm for arbitrary integer m. Self-dual codes of length 8 are also classified over Zpq for p,q=2,3,5,7.     

环F2+uF2+vF2上自对偶码的两种构造方法

本文研究环R=F2+uF2+vF2上的自对偶码问题.利用Rn到F3n2的Gray映射及R上的自对偶码C的Gray像为F2上自对偶码,获得了R上任何偶长度的自对偶码存在性的结论.最后,给出了R上两种构造自对偶码的方法.     

On Harmonic Weight Enumerators of Binary Codes

We define some new polynomials associated to a linear binary code and a harmonic function of degree k. The case k=0 is the usual weight enumerator of the code. When divided by (xy) ^k , they satisfy a MacWilliams type equality. When applied to certain harmonic functions constructed from Hahn polynomials, they can compute some information on the intersection numbers of the code. As an application, we classify the extremal even formally self-dual codes of length 12.     

Automorphisms of Constant Weight Codes and of Divisible Designs

We show that the automorphism group of a divisible design is isomorphic to a subgroup H of index 1 or 2 in the automorphism group of the associated constant weight code. Only in very special cases H is not the full automorphism group.     

New binary self-dual codes of lengths 50–60

Using a method for constructing binary self-dual codes with an automorphism of odd prime order \(p\) , we give a full classification of all optimal binary self-dual \([50+2t,25+t]\) codes having an automorphism of order 5 for \(t=0,\dots ,5\) . As a consequence, we determine the weight enumerators for which there is an optimal binary self-dual \([52, 26, 10]\) code. Some of the constructed codes for lengths 52, 54, 58, and 60 have new values for the parameter in their weight enumerator. We also construct more than 3,000 new doubly-even \([56,28,12]\) self-dual codes.     

On self-dual codes over {\mathbb{F}}_5

The purpose of this paper is to improve the upper bounds of the minimum distances of self-dual codes over for lengths [22, 26, 28, 32–40]. In particular, we prove that there is no [22, 11, 9] self-dual code over , whose existence was left open in 1982. We also show that both the Hamming weight enumerator and the Lee weight enumerator of a putative [24, 12, 10] self-dual code over are unique. Using the building-up construction, we show that there are exactly nine inequivalent optimal self-dual [18, 9, 7] codes over up to the monomial equivalence, and construct one new optimal self-dual [20, 10, 8] code over and at least 40 new inequivalent optimal self-dual [22, 11, 8] codes.