Binary Optimal Odd Formally Self-Dual Codes

In this paper, we study binary optimal odd formallyself-dual codes. All optimal odd formally self-dual codes areclassified for length up to 16. The highest minimum weight ofany odd formally self-dual codes of length up to 24 is determined. We also show that there is a unique linearcode for parameters [16, 8, 5] and [22, 11, 7], up to equivalence.     

On the Binary Self-Dual Codes with an Automorphism of Order 2

Methods to design binary self-dual codes with an automorphism of order two without fixed points are presented. New extremal self-dual [40,20,8], [42,21,8],[44,22,8] and [64,32,12] codes with previously not known weight enumerators are constructed.     

有限域上的重根自对偶负循环码

Let F_q be a finite field with q = p~m, where p is an odd prime. In this paper, we study the repeated-root self-dual negacyclic codes over Fq. The enumeration of such codes is investigated. We obtain all the self-dual negacyclic codes of length 2~ap~r over F_q, a ≥ 1.The construction of self-dual negacyclic codes of length 2~abp~r over F_q is also provided, where gcd(2, b) = gcd(b, p) = 1 and a ≥ 1.     

Classification of Extremal Double Circulant Self-Dual Codes of 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 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.     

Higher Weights for Ternary and Quaternary Self-Dual Codes*

We study higher weights applied to ternary and quaternary self-dual codes. We give lower bounds on the second higher weight and compute the second higher weights for optimal codes of length less than 24. We relate the joint weight enumerator with the higher weight enumerator and use this relationship to produce Gleason theorems. Graded rings of the higher weight enumerators are also determined. This work was supported in part by Northern Advancement Center for Science & Technology and the Natural Sciences and Engineering Research Council of Canada.     

Type II Self-Dual Codes over Finite Rings and Even Unimodular Lattices

In this paper, we investigate self-dual codes over finite rings, specifically the ring of integers modulo 2m. Type II codes over are introduced as self-dual codes with Euclidean weights which are a multiple of 2m +1. We describe a relationship between Type II codes and even unimodular lattices. This relationship provides much information on Type II codes. Double circulant Type II codes over are also studied.     

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.     

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.     

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.     

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.     

On the Automorphisms of Order 2 with Fixed Points for the Extremal Self-Dual Codes of Length 24"m

It is shown that an extremal self-dual code of length 24">m may have an automorphism of order 2 with fixed points only for ">m = 1,3, or 5. We prove that no self-dual [72, 36, 16] code has such an automorphism in its automorphism group.     

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     

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.     

Self-Orthogonal 3-(56,12,65) Designs and Extremal Doubly-Even Self-Dual Codes of Length 56

In this paper, we show that the code generated by the rows of a block-point incidence matrix of a self-orthogonal 3-(56,12,65) design is a doubly-even self-dual code of length 56. As a consequence, it is shown that an extremal doubly-even self-dual code of length 56 is generated by the codewords of minimum weight. We also demonstrate that there are more than one thousand inequivalent extremal doubly-even self-dual [56,28,12] codes. This result shows that there are more than one thousand non-isomorphic self-orthogonal 3-(56,12,65) designs. AMS Classification: 94B05, 05B05     

Extremal Self-Dual [50, 25, 10] Codes with Automorphisms of Order 3 and Quasi-Symmetric 2-(49, 9,6) Designs

Five non-isomorphic quasi-symmetric 2-(49, 9, 6) designs are known. They arise from extremal self-dual [50, 25, 10] codes with a certain weight enumerator. Four of them have an automorphism of order 3 fixing two points. In this paper, it is shown that there are exactly 48 inequivalent extremal self-dual [50, 25, 10] code with this weight enumerator and an automorphism of order 3 fixing two points. 44 new quasi-symmetric 2-(49, 9, 6) designs with an automorphism of order 3 are constructed from these codes.     

On Generalized Hamming Weights for Galois Ring Linear Codes

The definition of generalized Hamming weights (GHW) for linear codes over Galois rings is discussed. The properties of GHW for Galois ring linear codes are stated. Upper and existence bounds for GHW of – linear codes and a lower bound for GHW of the Kerdock code over – are derived. GHW of some – linear codes are determined.     

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 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.