A study of codes over some finite non chain rings

Abstract

The class of cyclic codes plays a significant role in the theory of error correcting codes. Cyclic codes can be efficiently encoded using simple shift registers. They have rich algebraic structures for efficient error detection and correction, which explains their preferred role in engineering. Quadratic residue codes is an important class of cyclic codes which have been introduced to construct self-dual codes. Quadratic residue codes have been generalized to duadic codes and to m-adic residue codes. Triadic codes generalize duadic codes, which have been further generalized to polyadic codes. Thus polyadic codes are the generalizations of quadratic residue codes, duadic codes, triadic codes and m-adic residue codes. Cyclic codes have been extended to negacyclic codes and then to constacyclic codes. newlineConstacyclic codes and polyadic codes over finite fields have been investigated by several authors. There has been recent development on codes over finite rings. Interest in these codes increased substantially after a break-through work by Hammons et al. A lot of research has been done in studying linear codes over finite rings such as, integer residue rings, Galois rings, chain rings and non-chain rings. In this thesis, we will study cyclic codes, constacyclic codes, quadratic residue codes, duadic codes, triadic codes, duadic negacyclic codes, polyadic cyclic codes and polyadic constacyclic codes over certain non-chain rings such as Fq + uFq + u2Fq + u3Fq with u4 = u and q is a prime congruent to 1 modulo 3, Fq + uFq + u2Fq + + um-1Fq, where um = u, m and#61619; 2 is any natural number and q is a prime power congruent to 1 modulo (m 1), ,( )[ ]f u F u q where f(u) is any polynomial of degree m, m and#61619; 2, which splits into distinct linear factors with Fq and , ( ), ( ),[ , ]f u g v uv vu F u v q and#61485;where f(u) and g(v) are polynomials, not both linear, which split into distinct linear factors over Fq.