Weight Functions and theExtension Theorem for Linear

An extension theorem for general weight functions is proved over nite chain 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 7], 8] proved an equivalence theorem: two linear codes C 1 ; C 2 F n deened over a nite eld F are equivalent up to monomial transformations if and only if there is a linear isomorphism f : C 1 ! C 2 which preserves Hamming weight. Bogart, Goldberg, and Gordon 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 the-oretic techniques to linear codes deened 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 deened over nite chain rings, i.e., nite commutative local principal ideal rings. Goldberg proved the extension theorem for symmetrized weight compositions over nite elds, 5], and Constan-tinescu, 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 This paper is in nal form and no version of it will be submitted for publication elsewhere. c 0000 (copyright holder)