Enumeration of finite commutative chain rings

Constructions of self-dual codes over finite commutative chain rings

We study self-dual codes over chain rings. We describe a technique for constructing new self-dual codes from existing codes and we prove that for chain rings containing an element c with c = −1 all self-dual codes can be constructed by this technique. We extend this construction to self-dual codes over principal ideal rings via the Chinese Remainder Theorem. We use torsion codes to describe the...

Finite Commutative Rings

‡ Every function on a finite residue class ring D/I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all a, b∈R, if ab∈ P then a∈ P k , b∈ P j with k+ j ≥min(n,N), we determine the number of functions (as...

Additive cyclic codes over finite commutative chain rings

Additive cyclic codes over Galois rings were investigated in [3]. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the study in [3], whereas the other one has some unusual propertie...

Commutative Rings and Finite Fields

Diophantine properties of finite commutative rings

Simple observations on diophantine definability over finite commutative rings lead to a characterization of those rings in terms of their diophantine behavior. A.M.S. Classification: 13M10, 11T06, 03G99.