The cocyclic Hadamard matrices of order less than 40

In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72.     

2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order,and their related Hadamard matrices and codes

We present the full classification of Hadamard 2-(31,15,7), Hadamard 2-(35, 17,8) and Menon 2-(36,15,6) designs with automorphisms of odd prime order. We also give partial classifications of such designs with automorphisms of order 2. These classifications lead to related Hadamard matrices and self-dual codes. We found 76166 Hadamard matrices of order 32 and 38332 Hadamard matrices of order 36, arising from the classified designs. Remarkably, all constructed Hadamard matrices of order 36 are Hadamard equivalent to a regular Hadamard matrix. From our constructed designs, we obtained 37352 doubly-even [72,36,12] codes, which are the best known self-dual codes of this length until now.      

Skew Hadamard designs and their codes

Skew Hadamard designs (4n – 1, 2n – 1, n – 1) are associated to order 4n skew Hadamard matrices in the natural way. We study the codes spanned by their incidence matrices A and by I + A and show that they are self-dual after extension (resp. extension and augmentation) over fields of characteristic dividing n. Quadratic Residues codes are obtained in the case of the Paley matrix. Results on the p-rank of skew Hadamard designs are rederived in that way. Codes from skew Hadamard designs are classified. An optimal self-dual code over GF(5) is rediscovered in length 20. Six new inequivalent [56, 28, 16] self-dual codes over GF(7) are obtained from skew Hadamard matrices of order 56, improving the only known quadratic double circulant code of length 56 over GF(7).     

Classification of Hadamard matrices of order 44 with automorphisms of order 7

The Hadamard matrices of order 44 possessing automorphisms of order 7 are classified. The number of their equivalence classes is 384. The order of their full automorphism group is calculated. These Hadamard matrices yield 1683 nonisomorphic 3-(44,22,10) designs, 57932 nonisomorphic 2-(43,21,10) designs, and two inequivalent extremal binary self-dual doubly even codes of length 88 (one of them being new).     

Orthogonal Designs and Type II Codes over ℤ2k

Symmetric designs are used to construct binary extremal self-dual codes and Hadamard matrices and weighing matrices are used to construct extremal ternary self-dual codes. In this paper, we consider orthogonal designs and related matrices to construct self-dual codes over a larger alphabet. As an example, a number of extremal Type II codes over _2k are constructed.     

Cocyclic Development of Designs

We present the basic theory of cocyclic development of designs, in which group development over a finite group G is modified by the action of a cocycle defined on G × G. Negacyclic and -cyclic development are both special cases of cocyclic development.Techniques of design construction using the group ring, arising from difference set methods, also apply to cocyclic designs. Important classes of Hadamard matrices and generalized weighing matrices are cocyclic.We derive a characterization of cocyclic development which allows us to generate all matrices which are cocyclic over G. Any cocyclic matrix is equivalent to one obtained by entrywise action of an asymmetric matrix and a symmetric matrix on a G-developed matrix. The symmetric matrix is a Kronecker product of back -cyclic matrices, and the asymmetric matrix is determined by the second integral homology group of G. We believe this link between combinatorial design theory and low-dimensional group cohomology leads to (i) a new way to generate combinatorial designs; (ii) a better understanding of the structure of some known designs; and (iii) a better understanding of known construction techniques.     

Orbit matrices of Hadamard matrices and related codes

In this paper we introduce the notion of orbit matrices of Hadamard matrices with respect to their permutation automorphism groups and show that under certain conditions these orbit matrices yield self-orthogonal codes. As a case study, we construct codes from orbit matrices of some Paley type I and Paley type II Hadamard matrices. In addition, we construct four new symmetric (100,45,20) designs which correspond to regular Hadamard matrices, and construct codes from their orbit matrices. The codes constructed include optimal, near-optimal self-orthogonal and self-dual codes, over finite fields and over ${\mathbb{Z}}_4$.     

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.     

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     

On the classification and enumeration of self-dual codes

A complete classification is given of all [22, 11] and [24, 12] binary self-dual codes. For each code we give the order of its group, the number of codes equivalent to it, and its weight distribution. There is a unique [24, 12, 6] self-dual code. Several theorems on the enumeration of self-dual codes are used, including formulas for the number of such codes with minimum distance ? 4, and for the sum of the weight enumerators of all such codes of length n. Selforthogonal codes which are generated by code words of weight 4 are completely characterized.     

Hadamard matrices of order 36 formed by codewords in some ternary self-dual codes

In this note, we study the existence of Hadamard matrices of order 36 formed by codewords of weight 36 in some ternary near-extremal self-dual codes of length 36.     

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.     

Codes with given distances

One of the main results says that ifC is a binary linear code of length 4t and of dimension greater than 2t, thenC contains a word of weight 2t and this bound is best possible. Several results of similar flavor are established both for linear and non-linear codes. For the proof a lemma introducing the binormal forms of binary matrices is needed. The results are applied to determine the coset chromatic number of Hadamard graphs, to solve a problem of Galvin and to give a short proof of a theorem of Gleason on self-dual doubly-even codes.     

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.      

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.      

Trellis Structure and Higher Weights of Extremal Self-Dual Codes

A method for demonstrating and enumerating uniformly efficient (permutation-optimal) trellis decoders for self-dual codes of high minimum distance is developed. Such decoders and corresponding permutations are known for relatively few codes.The task of finding such permutations is shown to be substantially simplifiable in the case of self-dual codes in general, and for self-dual codes of sufficiently high minimum distance it is shown that it is frequently possible to deduce the existence of these permutations directly from the parameters of the code.A new and tighter link between generalized Hamming weights and trellis representations is demonstrated: for some self-dual codes, knowledge of one of the generalized Hamming weights is sufficient to determine the entire optimal state complexity profile.These results are used to characterize the permutation-optimal trellises and generalized Hamming weights for all [32,16,8] binary self-dual codes and for several other codes. The numbers of uniformly efficient permutations for several codes, including the [24,12,8] Golay code and both [24,12,9] ternary self-dual codes, are found.     

The extremal codes of lengths 76 with an automorphism of order 19

We find all extremal [76,38,14] binary self-dual codes having automorphism of order 19. There are three inequivalent such codes. One of them was previously known. The other two are new. These codes are the shortest known self-dual codes of minimal weight 14 as well as the best-known linear codes of that length and dimension.     

A Mixed Heuristic for Generating Cocyclic Hadamard Matrices

A way of generating cocyclic Hadamard matrices is described, which combines a new heuristic, coming from a novel notion of fitness, and a peculiar local search, defined as a constraint satisfaction problem. Calculations support the idea that finding a cocyclic Hadamard matrix of order \(4 \cdot 47\) might be within reach, for the first time, progressing further upon the ideas explained in this work.