Jay A. Wood

DUALITY FOR MODULES OVER FINITE RINGS AND APPLICATIONS TO CODING THEORY

This paper sets a foundation for the study of linear codes over finite rings. The finite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from finite fields to finite Frobenius rings. It is over Frobenius rings that certain key identifications can be made between the ring and its complex characters.

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.

The structure of linear codes of constant weight

In this paper we determine completely the structure of linear codes over Z/NZ of constant weight. Namely, we determine exactly which modules underlie linear codes of constant weight, and we describe the coordinate functionals involved. The weight functions considered are: Hamming weight, Lee weight, two forms of Euclidean weight, and pre-homogeneous weights. We prove a general uniqueness theorem for virtual linear codes of constant weight. Existence is settled on a case by case basis.

Code equivalence characterizes finite Frobenius rings

In this paper we show that finite rings for which the code equivalence theorem of MacWilliams is valid for Hamming weight must necessarily be Frobenius. This result makes use of a strategy of Dinh and Lopez-Permouth.

Extension Theorems for Linear Codes over Finite Rings

Various forms of the extension problem are discussed for linear codes defined over finite rings. The extension theorem for symmetrized weight compositions over finite Frobenius rings is proved. As a consequence, an extension theorem for weight functions over certain finite commutative rings is also proved. The proofs make use of the linear independence of characters as well as the linear independence of characters averaged over the orbits of a group action.

FOUNDATIONS OF LINEAR CODES DEFINED OVER FINITE MODULES: THE EXTENSION THEOREM AND THE MACWILLIAMS IDENTITIES

This paper discusses the foundations of the theory of linear codes defined over finite modules. Two topics are examined in depth: the extension theorem and the MacWilliams identities. Both of these topics were studied originally by MacWilliams in the context of linear codes defined over finite fields.

Spinor groups and algebraic coding theory

Abstract The purpose of this paper is to explore the equivalence between the abelian subgroups of Ṽ(n), the “diagonal” extra-special 2-group of the compact, simple, simply-connected Lie group Spin(n), and the self-orthogonal linear binary codes of algebraic coding theory. In particular, the basic abstract structure theory of the abelian subgroups of Ṽ(n) is reflected in the distinction between ordinary self-orthogonal codes and even self-orthogonal codes. Work of Quillen on the equivariant cohomology of Spin(n) affects the classification of even self-orthogonal codes.

Witt’s extension theorem for mod four valued quadratic forms

The mod 4 valued quadratic forms defined by E. H. Brown, Jr. are studied. A classification theorem is proven which states that these forms are determined by two things: whether or not their associated bilinear form is alternating, and the rj-invariant of Brown (which generalizes the Arf invariant of an ordinary quadratic form). Particular attention is paid to a generalization of Witts extension theorem for quadratic forms. Some applications to self- orthogonal codes are sketched, and an exposition of some unpublished work of E. Prange on Witts theorem is provided in an appendix.

MacWilliams' Extension Theorem for bi-invariant weights over finite principal ideal rings

A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius. This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property. Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal rings, which form a large subclass of the finite Frobenius rings. We do not assume commutativity of the rings in question.

Relative one-weight linear codes

Relative one-weight linear codes were introduced by Liu and Chen over finite fields. These codes can be defined just as simply for egalitarian and homogeneous weights over Frobenius bimodule alphabets. A key lemma helps describe the structure of relative one-weight codes, and certain known types of two-weight linear codes can then be constructed easily. The key lemma also provides another approach to the MacWilliams extension theorem.