Talk abstract:

Duals over Artinian Rings and the MacWilliams Identity

Thomas Mittelholzer
IBM Zurich Research Laboratory
Switzerland
tmi@zurich.ibm.com

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 Rⁿ. 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. In the case of arbitrary (infinite) fields, a MacWilliams-like identity is derived, which relates the dimensions of certain Hamming distance related subspaces of a code and its dual.

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