Linear Codes and Their Duals Over Artinian Rings

Abstract

Linear codes over commutative artinian rings R are considered. For a linear functional-based definition of duality, it is shown that the class of length-n linear block codes over R should consist of projective submodules of the free module R n. For this class, the familiar duality properties from the field case can be generalized to the ring case. In particular, the MacWilliams identity is derived for linear codes over any finite commutative ring. Duals of convolutional codes are also considered, and it is shown that for convolutional codes over commutative artinian rings, the duality property holds for a code and its dual as well as for the local description of the code by its canonical trellis section and its dual trellis section.