ON CYCLIC CODES AND DUADIC CODES OVER SOME FINITE RINGS

Abstract

Coding theory deals with the construction and analysis of error-correcting codes for the reliable and efficient transmission of information through noisy communication channels. Algebraic coding theory is that branch of coding theory that mainly uses algebraic tech niques for the construction and analysis of codes. Initially algebraic codes had been stud ied as vector spaces over finite fields. In early seventies, the study of algebraic codes over finite rings was initiated [24,25,103,104,119,127]. However, a serious study in this area started only after the research paper of Hammons et al. [60], where they have shown that some non-linear binary codes can be obtained as Gray images of some linear codes over Z4. After this work, researchers took a keen interest on codes over rings, and sev eral families of codes over rings have been studied [21,28,31,32,34,37,58,70,87,129]. In particular, codes over finite chain rings like Zm for some positive integer m, and Galois rings [2,3,26,46,48,88,92,93,114–116,122,125] were studied. However, little attention was paid to the study of codes over non-chain rings. Recently, many researchers have studied codes over some non-chain rings such as F2+vF2, v2 = v [138], Fp+uFp+vFp+uvFp, u2 = 0, v2 = 0,uv = vu, where p is a prime [15–17,61,97,132–134]. Yildiz and Karadeniz [131] studied linear and self-dual codes over a local non-chain ring Z4 +uZ4,u2 = 0, and, Gao et al. [55] have discussed cyclic codes over the ring Zq + uZq,u2 = 0, where q = pr and p is a prime, which is a local non-chain ring. Some codes with better parameters than comparable codes over finite fields have been obtained over such rings. In this thesis, we have studied some classes of codes over some non-chain rings which are extensions of the rings Zq and Fp, where p is a prime and q is a prime power. The family of cyclic codes is an important family of algebraic codes. Many authors have studied these codes over finite chain rings [1,2,4,26,48,88]. In this context, we have studied cyclic i ii codes of length n over Zq + uZq, u2 = 0, where q = pr, p a prime and (n,p) = 1. We have obtained the complete ideal structure of Zq+uZq and then using this, the structure of cyclic codes and that of their duals is obtained through the factorization of xn −1 over Zq +uZq. We have enumerated the total number of cyclic codes of length n over Zq +uZq. Self-dual codes are an important class of linear codes. We have presented a necessary and sufficient condition for a cyclic code of length n over Zq + uZq to be self-dual. We have computed the total number of self-dual cyclic codes of length n over Zq + uZq. A new Gray map from Zq + uZq to Z2r p is introduced and some cyclic codes over Z4 with good parameters are obtained. In polynomial representation, a cyclic code of length n over a ring R is an ideal of the residue class ring R[x]/xn − 1 . If (n,p)= 1, then polynomial xn − 1 may have repeated roots in Zq + uZq . In this condition, cyclic code of length n over Zq + uZq is called a repeated root cyclic code. We have studied repeated root cyclic codes of length pk over Zp2 + uZp2, u2 = 0. A unique set of generators for these codes is presented and an upper bound for the Lee distance of linear codes over Zp2 + uZp2 is obtained, and with respect to this bound, some optimal repeated root cyclic codes are obtained. We have obtained a minimal spanning set for cyclic codes of such type and determined their ranks. In addition, we have determined the complete algebraic structure of principally generated cyclic codes in this class. Constacyclic codes are an important generalization of cyclic codes. The family of con stacyclic codes contains codes with better parameters compared to the family of cyclic codes. We have studied (1 + 2s−1u)-constacyclic and skew (1 + 2s−1u)-constacyclic codes of odd length over the ring Z2s + uZ2s, u2 = 0, where s ≥ 3 is an odd integer. We have obtained the algebraic structure of (1 + 2s−1u)-constacyclic codes over Z2s + uZ2s, and by defining some new Gray maps from Z2s+uZ2s to Z2+uZ2, it is shown that the Gray images of (1 +2s−1u)-constacyclic and skew (1 + 2s−1u)-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over Z2 + uZ2. An important generalization of cyclic codes is the family of abelian codes. Abelian codes over finite fields were introduced by Berman [20] and MacWilliams [85]. After these works, these codes have been studied by many authors [99,100,118]. Duadic codes are a iii class of abelian codes. We have studied duadic codes of odd length over Z4 +uZ4 and over F2 + uF2 +vF2 +uvF2. We have studied these codes by considering them as a class of abelian codes and using the Fourier transform approach. We have studied some properties of the torsion and residue codes of abelian codes. Some results related to self-duality and self-orthogonality of duadic codes are presented. Some conditions on the existence of self dual augmented and extended codes over Z4+uZ4 as well as over F2+uF2+vF2+uvF2 are determined. We have presented a sufficient condition for abelian codes of same length over these rings to have the same minimum Hamming distance. A new Gray map over Z4+uZ4 is defined, and it is shown that the Gray image of an abelian code over Z4+uZ4 is an abelian code over Z4. We have obtained five new linear codes of length 18 over Z4 from duadic codes of length 9 over Z4 +uZ4 as images of Gray map and a new map defined from Z4+uZ4 to Z2 4. The parameters of these codes are [18,44210,4],[18,4528,4],[18,4425,8],[18,4029,8] and [18, 4225, 6]. The code with parameters [18,4029,8] is self-orthogonal. These codes have been reported and added to the database of Z4-codes [7]. Further, we have shown that the Gray image of an abelian code of length n over F2 + uF2 + vF2 + uvF2 is a binary abelian code of length 4n. We have also studied some conditions for self-dual augmented and extended codes over F2 +uF2 +vF2 +uvF2 to be Type II codes. Some results related to the minimum Hamming and Lee distances of duadic codes over F2 + uF2 + vF2 +uvF2 are presented. Some optimal binary linear codes are obtained as Gray images of abelian codes over F2 +uF2 +vF2 +uvF2.