Self-Dual Codes and Modules for Finite Groups in Characteristic Two

In this chapter, we describe self-dual codes over Frobenius rings. We give constructions of self-dual codes over any Frobenius ring. We describe connections to unimodular lattices, binary self-dual codes and to designs. We also describe linear complementary dual codes and make a new definition of a broad generalization encompassing both self-dual and linear complementary codes.

Binary self-dual codes and additive self-dual codes over GF(4) contain common points. Both have Type I codes and Type II codes, as well as shadow codes. In this paper, we provide a comprehensive description of extremal and near-extremal Type I codes over GF(2) and GF(4) with minimal shadow. In particular, we prove that there is no near-extremal Type I [24m,12m,2m+2] binary self-dual code with minimal shadow if m≥323, and we prove that there is no near-extremal Type I (6m+1,26m+1,2m+1) additive self-dual code over GF(4) with minimal shadow if m≥22.

In this work, we investigate the second generalized Hamming weights for binary doubly-even self-dual codes from the point of view of corresponding t-designs by the Assmus-Mattson theorem. In particular, for extremal doubly-even self-dual codes, we shall give a bound on the weights and determine the weights by using the block intersection numbers of corresponding t-designs. Moreover we study the support weight enumerators for binary doubly-even self-dual codes and determine the second support weight enumerators for binary extremal doubly-even self-dual codes of length 56 and 96.

In this chapter, we shall present some fundamental study on relatively self-dual codes over a finite commutative ring. Section 1.1 is devoted to the basic definitions and properties of such codes. In Section 1.2, we shall present our gluing technique for constructing relatively self-dual codes. Several exotic general constructions of self-dual codes over ℤn will be given in Section 1.3. Four constructions of binary self-dual codes by our gluing technique will be given in Section 1.4. In Section 1.5, we shall give some constructional examples of ternary self-dual codes. In Section 1.6, we shall present some families of self-dual codes over ℤ4. Sections 1.1, 1.2 and Constructions III, IV in Section 1.4 were published in [X2].

In this work, we study codes over the ring 𝓡 k, m = 𝔽2[u, v]/ 〈uk , vm , uv − vu〉, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from 𝓡 k, m to F 2 k m $\mathbb{F}_2^{km}$ together with a Lee weight. After proving the MacWilliams identities for codes over 𝓡 k, m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over 𝓡 k, m . In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72, 36, 12] and 105 new Type II binary self-dual codes of parameter [72, 36, 12].

It is shown that if the automorphism group of a binary self-dual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4 . In particular, no cyclic binary self-dual code can have all its weights divisible by four. The number of cyclic binary self-dual codes of length n is determined, and the shortest nontrivial code in this class is shown to have length 14 .

We give a 1-level squaring construction for binary repeated-root cyclic codes of length n=2/sup a/b, a/spl ges/1, b odd. This allows us to obtain the weight distributions of all cyclic binary self-dual codes of lengths up to 110, which are not accessible by direct computation. We also use the shadow construction, as a particular method for Type I self-dual codes.

For a binary linear code ${\cal C}$ and a vector u, we define and study the overlap polynomial $F^{(u)}_{{\cal C}}(Z,X,D)$ which is a sum over the words in ${\cal C}$ of terms that takes into account the weight of the element of ${\cal C}$ and its overlap with the vector $u$. We obtain a generalized MacWilliams relation between $F^{(u)}_{{\cal C}}(Z,X,D)$ and $F^{(u)}_{{\cal C}^{\perp}}(Z,X,D)$ and a recursion equation for determining the overlap polynomials of a self-dual code in terms of overlap polynomials corresponding to vectors of lower weight. We apply the recursion to determine the weight enumerators of the cosets of the (putative) [72,36,16] self-dual code of Type II, up to a small number of parameters. The overlap polynomials are expressed in terms of covering polynomials $g^{(u)}_{{\cal C}}(Z,X)$ that are sums over the elements of ${\cal C}$ that cover $u$. The design strength $\delta({\cal C})$ is closely related to the covering polynomials and we use them to get a sharpened version of the Assmus--Mattson theorem. In many cases we are able to conclude that certain codes hold $\delta({\cal C})$-designs but no weight class holds a $1+\delta({\cal C})$-design. We study the design strength of self-dual, doubly even extremal codes and, in the case of length divisible by 24, improve the well-known estimate $\delta({\cal C})\geq 5$ to state that either $\delta({\cal C})\geq 7$ or $\delta({\cal C})=5$ and no weight class holds a 6-design; similar results hold in the case of other lengths. Examples are given using a self-dual code [32,16,8], certain cyclic subcodes of the punctured Reed--Muller code ${\cal R}(2,m)^*$, the first order RM-code, and a [16,9,4] code in which all six of the weight classes hold 2-designs but one nontrivial weight class also holds a 3-design.

A divisible code is a linear code whose word weights have a common divisor larger than one. If the divisor is a power of the field characteristic, there is a simple bound on the dimension of the code in terms of its weight range. When this bound is applied to the subcode of words with weight divisible by four in a type I binary self-dual code. It yields an asymptotic improvement of the Conway-Sloane bound for self-dual codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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

It is shown that (1) every almost self-injective group algebra is self-injective and (2) if the group algebra KG is continuous, then G is a locally finite group. Furthermore, it follows that the following assertions are equivalent: a CS group algebra KG is continuous; KG is principally self-injective; the group G is locally finite.

We classify, in terms of the structure of the finite group G, all group algebras KG for which all right ideals are right annihilators of principal left ideals. This means in the language of coding theory that we classify code-checkable group algebras KG which have been considered so far only for abelian groups G. Optimality of checkable codes and asymptotic results are discussed.

We study self-dual codes (SD) over GF(2) with 8 cycles of length 11 in the decomposition of its automorphism and minimum weight at least 16. The main method we use is by Huffman and Yorgov and is for constructing SD codes having an automorphism of order p for a prime p > 2. We prove that [8],[4] SD codes over F210 such that their preimage is a binary code with minimum weight 20 does not exist. By restricting the matrices of the subcode to have symmetric or circulant submatrix in their systematic form, we have found many new Hermitian self-dual codes over F210. Those newly found Hermitian codes allow us to constructed 468062 new binary SD [88,44,16] doubly-even codes.

Binary cyclic self-dual codes of odd length do not exist. It is not the case, however, of Z/sub 4/-cyclic self-dual codes. Some of the first examples, aside from the trivial self-dual code, are supplemented quadratic residue codes. The aim of this paper is to characterize arithmetically the (odd) lengths where such non-trivial cyclic self-dual Z/sub 4/-codes can exist and to give some examples for short lengths. As the length is odd, it is hopeless to try to obtain directly Type II codes; i.e. codes whose Euclidean weights are multiples of 8 since these exist only for lengths a multiples of 8. It is possible, nonetheless, to obtain Type I Z/sub 4/-codes, and, accordingly, by construction A, Type I lattices. We obtain, in that way, the only two extremal odd lattices in dimensions 15-47: the shorter Leech lattice O/sub 23/ in dimension 23 and A/sub 15//sup +/, in dimension 15. Invariant theory enables us to compute the symmetric weight enumerators of the codes and therefore the theta series of the associated lattices, including the norm and kissing number thereof. In passing, we mention an amusing non-existence arithmetic criterion for cyclic projective planes.

We introduce linear cyclic codes over the ring F/sub 2/+uF/sub 2/={0,1,u,u~=u+1}, where u/sup 2/=0 and study them by analogy with the Z/sub 4/ case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z/sub 4/-codes. Unlike the Z/sub 4/ case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-dual codes through a linear Gray map.