Semigroup Rings and the Extension Theorem for Linear Codes

An extension theorem for general weight functions is proved over nite commutative local principal ideal rings. The structure of the complex semigroup ring associated to the multiplicative semigroup of the ring plays a prominent role in the proof. 1. Background In her doctoral dissertation, MacWilliams [8], [9] proved an equivalence theorem: two linear codes C1; C2 F n de ned over a nite eld F are equivalent up to monomial transformations if and only if there is a linear isomorphism f : C1 ! C2 which preserves Hamming weight. Bogart et al. [2] gave another proof of this theorem, and a character theoretic proof was provided by Ward and the author [13]. Following up on the ideas in [13], the author has extended the character theoretic techniques to linear codes de ned over nite Frobenius rings, rst for the Hamming weight [15] and then for symmetrized weight compositions [16]. In this paper, the author treats general weight functions de ned over nite commutative local principal ideal rings. Goldberg proved the extension theorem for symmetrized weight compositions over nite elds, [6], and Constantinescu, Heise, and Honold have proved an extension theorem for homogeneous weight functions over Z=m, [4]. A word on the name of the theorem. MacWilliams' result above is sometimes referred to as \the equivalence theorem of MacWilliams." I have come to prefer \the extension theorem of MacWilliams," because of the similarity to the extension theorems of Witt [14] and Arf [1] for bilinear and quadratic forms. In all these situations there is a xed ambient space V , usually a nite dimensional vector space over a eld. The space V is equipped with an auxiliary function, a weight function in coding theory, a bilinear or quadratic form otherwise. The linear automorphisms of V which preserve the auxiliary function form a group of linear isometries, often a classical group in the case of bilinear or quadratic forms, often a group of monomial transformations in coding theory. The extension theorem then determines conditions under which any injective linear transformation f : W ! V from a subspace W of V which preserves the auxiliary function must in fact extend to a linear isometry of V itself. 2. Statement of the extension problem Fix a nite associative ring R with 1. (Later, we will impose additional hypotheses on R, but we will try to be as general as possible for as long as possible.) Let R denote the free module consisting of n-tuples of elements from R. A right linear code of length n is a right submodule C R. The complete weight composition is the function Partially supported by NSA grants MDA904-94-H-2025 and MDA904-96-1-0067, and by Purdue University Calumet Scholarly Research Awards.