Linear Block Codes over the Non Chain Ring Fp+ vFp + v2Fp

Abstract

Codes over finite rings have been studied in the early 1970. A great deal of attention has been given to codes over finite rings from 1990, because of their new role in algebraic coding theory and their successful application. The key to describing the structure of cyclic codes over a ring R´ is to view cyclic codes as ideals in the polynomial ring R´ [x] /< xn -1>, where n is the length of the code. In this thesis we study the structure and properties of linear and cyclic codes over the ring R2 = F2 + vF2 + v2F2 where v3 = v, which is a semi-local not chain ring. That we first study the relationship between cyclic codes over F2 + vF2 + v2F2 and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the generator polynomial of cyclic codes over the ring, and we obtain the unique idempotent generators for cyclic codes of odd length and we study the (1 + v + v2)-constacyclic codes over R2. We also extend the study to linear and cyclic codes over the ring Rp = Fp + vFp + v2Fp, where v3 = v and p is prime greater than 2 such that Rp is a semi-local not chain ring. We firstly give the generator matrix of linear codes and their dual codes over Rp. Then, we define a Gray map Ψ from Rpn to Fp3n, and obtain Gray images Ψ(C) from the generator matrix of linear codes C over the ring Rp and we investigate the structure of cyclic codes over the ring Rp and give generator polynomials of cyclic codes over this ring . Then we study constacyclic codes over Rp. That we characterize the polynomial generators of all constacyclic codes over Rp, and show that constacyclic codes over Rp of arbitrary length are principally generated and we also discuss dual codes of constacyclic codes over Rp. Finally we introduce and study quadratic residue codes over the ring Rp in terms of their idempotent generators. And we study the structure of these codes and observe that these codes share similar properties with quadratic residue codes over finite fields.