m-ADIC RESIDUE CODES OVER THE RING Fq[v]/(vs - v) AND THEIR APPLICATIONS TO QUANTUM CODES

Abstract

Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The m-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the m-adic residue codes over the quotient ring F-q[v]/ . We determine the idempotent generators of the m-adic residue codes over F-q[v]/ . We obtain some parameters of optimal m-adic residue codes over F-q[v]/ with respect to Griesmer bound for rings. Furthermore, we derive a condition for m-adic residue codes over F-q [v]/ to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing m-adic residue codes over F-q[v]/ and give some examples to illustrate our findings.