Semigroup Rings And The Extension Theorem For Linear Codes

. An extension theorem for general weight functions is proved over finite 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 C 1 ; C 2 ae F n defined over a finite field 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 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 defined over finite Frobenius rings, first for the Hamming weight [15] and then for symmetrized weight compositions [16]. In this paper, the author treats general weight functions defined over finite commutat..