Harold N. Ward

Self-dual codes over GF(4)

Abstract This paper studies codes C over GF(4) which have even weights and have the same weight distribution as the dual code C⊥. Some of the results are as follows. All such codes satisfy C ⊥ = C (If C⊥= C, T has a binary basis.) The number of such Cs is determined, and those of length ⩽14 are completely classified. The weight enumerator of C is characterized and an upper bound obtained on the minimum distance. Necessary and sufficient conditions are given for C to be extended cyclic. Two new 5-designs are constructed. A generator matrix for C can be taken to have the form [I | B], where B ⊥ = B . We enumerate and classify all circulant matrices B with this property. A number of open problems are listed.

Ternary codes of minimum weight 6 and the classification of the self-dual codes of length 20

Self-orthogonal ternary codes of minimum weight 3 may be analyzed in a straightforward manner using the theory of glueing introduced in earlier papers. The present paper describes a method for studying codes of minimum weight 6 : the supports of the words of weight 6 form what is called a center set. Associated with each center set is a graph, and all the graphs that can arise in this way are known. These techniques are used to classify the ternary self-dual codes of length 20 : there are 24 inequivalent codes, 17 of which are indecomposable. Six of the codes have minimum weight 6 .

Characters and the equivalence of codes

MacWilliams proved that two linear codes are equivalent up to monomial transformations if and only if there is a weight-preserving linear isomorphism between them. We present a new proof based on the linear independence of group characters. We also present a new proof of a theorem of Bonisoli characterizing constant weight codes.

Divisibility of Codes Meeting the Griesmer Bound

We prove that if a linear code overGF(p),pa prime, meets the Griesmer bound, then ifpedivides the minimum weight,pedivides all word weights. We present some illustrative applications of this result.

Quadratic residue codes and symplectic groups

The extended quadratic residue codes are known to be invariant under a monomial action by the projective special linear group, an action whose permutation part is the ordinary action on the projective line [3, Theorem 3.11 (see also [7], [15], [16]). Th e re p resentations of the special linear group that arise from the codes are among those constructed in [20] (the group coinciding with the symplectic group in the two-dimensional case). It was thought that the algebraic framework used in [20] to produce these representations could also serve as a starting-point for the codes, thus giving the codes and the group action at the same time. The purpose of the present paper is to substantiate that thought. Here is an outline of the paper and its main results: let V be a vector space of even dimension 2n over the finite field GF(p), 4 a power of an odd prime, and let V be endowed with a non-degenerate symplectic (skewsymmetric) form. Let Q(V) be the corresponding symplectic group on V. An algebra A was constructed in [20] that is basically a twisted group algebra of V over a suitable ring, and from A certain representations of Sp( V) were obtained. Section 1 summarizes these results. In Section 2 several functions arising in that development are calculated more explicitly. Their values involve the constant p of 2.1: p2 = 6q, with Q E 6 (mod 4), 6 = 51. To each maximal isotropic subspace of V corresponds a distinct idempotent of A, and the set I of these idempotents is central to the formation of the codes. The multiplication of these idempotents is also considered in Section 2, the most frequently used result being the Lemma 2.3. (2.6 deals with the characters of the representations.) If the symplectic form is scaled by a non-square of GF@), one obtains another algebra A’, and its relation to A is the subject of Section 3. Section 4 contains the construction of the codes and a monomial action

A restriction on the weight enumerator of a self-dual code

Abstract Gleasons theorem gives the general form of the weight enumerator of a linear binary self-dual code; it is a linear combination with integral coefficients of certain polynomials. When the subcode of words whose weights are multiples of 4 is not the whole code, the MacWilliams identities applied to that subcode yield divisibility conditions on those coefficients. The conditions show that there are no further extremal codes, for Case 1 in the sense of Mallows and Sloane, than the ones known.

A bound for divisible codes

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. >

Gold and Kasami-Welch functions, quadratic forms, and bent functions

We use elementary facts about quadratic forms in characteristic 2 to evaluate the sign of some Walsh transforms in terms of a Jacobi symbol. These results are applied to the Walsh transforms of the Gold and Kasami-Welch functions. We prove that the Gold functions yield bent functions when restricted to certain hyperplanes. We also use the sign information to determine the dual bent function.

Weight polarization and divisibility

Abstract A test for a code to be divisible, applicable to a spanning set, is developed from a formula for the polarization of the weight function.

A geometric approach to classifying Griesmer codes

We develop a geometric framework for the determination of codes meeting the Griesmer bound that involves a classification of ((q + 1)x, x) generalized minihypers in PG(2, q). These are multisets: each point of the plane has a multiplicity, and the strength of a line is the sum of its point multiplicities. The minihyper requirement is that each line has strength at least x, with some line having strength exactly x. When