Variations on minimal codewords in linear codes

Abstract

We address two topics related to the concept of minimal supports (codewords) in linear codes. In the first part we study the distribution of the number of minimal supports in random codes. In the second part, we propose a generalization of this concept for codes defined as modules over Galois rings. We determine minimal supports for some ℤ4-linear codes. Finally, we extend a recently established link between the cryptographical problem of secret sharing and minimal supports to the case of rings. The resulting secret-sharing schemes have fully and partially authorized coalitions, which permits, e.g., hierarchical access to a common resource.