ON CODES OVER SOME NON-CHAIN EXTENSIONS OF Z4

Abstract

Algebraic codes have been traditionally studied in the setting of vector spaces over nite elds. The study of codes over rings was initiated in early seventies [22,23,101,102,113,114, 122]. However, codes over rings got attention of researchers mainly after the breakthrough paper of Hammons et al. [56] in 1994, in which they have shown that some non-linear binary codes are actually the images of some linear codes over Z4 under the Gray map. The ndings of this paper led to a lot of research in this area [19,24,29{31,33,54,65,86,125]. As a result, a large number of papers were produced studying codes over Z4. This quickly expanded to consider codes over Zm and then onto other rings such as Galois rings and, in general, nite chain rings [3, 4, 20, 43, 47, 87, 93, 94, 120]. However, not much attention has been paid to codes over non-chain rings. Recently, nite polynomial rings such as F2 + uF2, u2 = 0 [25, 116], F2 + vF2, v2 = v [130], F2 + uF2 + vF2 + uvF2, u2 = v2 = 0, uv = vu [128], etc., have been considered as alphabets for studying codes. Some of these are non-chain rings. Recently, Yildiz and Karadeniz [129] have introduced a local non-chain ring Z4 + uZ4, u2 = 0, and studied linear and self-dual codes over it. They have also got some good formally self-dual codes over this ring. The study of codes over nite rings has provided some codes with better parameters and such codes have also got some practical applications. In this thesis, we have explored some families of codes over some non-chain extensions of Z4. In this context, we have introduced two new rings Z4 + vZ4, v2 = v, and Z4 + vZ4, w2 = 2w, and studied linear codes over them. Further, we have done an indepth study of cyclic and negacyclic codes over Z4 + uZ4, u2 = 0. Finding a suitable metric for the codes over a given ring is an interesting problem. In view of this, we have studied codes over Z4 + vZ4 with respect to Lee and Gray metrics and also derived MacWillaims type i identities for linear codes with respect to these metrics. A non-Hamming metric, namely, Rosenbloom-Tsfasman (RT) metric or the metric has also been considered in this study, and linear codes over Z4+vZ4 with respect to this metric are studied. A MacWilliams type identity using Lee complete -weight enumerator is presented. A transformation to obtain -weight enumerator from Lee complete -weight enumerator is provided. Self-dual codes are an interesting class of codes and are closely related to design theory. The study of self-dual codes and their constructions is an important topic in coding theory. We have characterized self-dual codes over Z4 + vZ4, v2 = v and Z4 + wZ4, w2 = 2w and presented some methods for constructing self-dual and self-orthogonal codes over Z4 +vZ4 and Z4+wZ4. We have also brie y studied circulant codes and Type II codes over Z4+wZ4. One of the most studied families of algebraic codes is the family of cyclic codes, which have a rich algebraic structure. Their structure over nite chain rings is now well known [43, 80, 81, 87]. However, they have not been well explored over local non-chain rings. We have studied cyclic codes over the non-chain ring Z4+uZ4, u2 = 0. First we have focused on cyclic codes of odd length n, and obtained their structure through the factorization of xnð€€€1 over Z4 + uZ4. We have then considered cyclic codes of arbitrary lengths over Z4 + uZ4 and presented their structure. In particular, all cyclic codes of length 2k are classi ed. Using the structure of general form of cyclic codes over Z4 + uZ4, we have obtained a minimal spanning set and a formula for the ranks of such codes. A necessary condition and a su cient condition for cyclic codes to be free over Z4 + uZ4 are obtained. An important generalization of cyclic codes is negacyclic codes. We have characterized negacyclic codes of both odd and even lengths over Z4 + uZ4, u2 = 0. The complete classi cation of negacyclic codes of length 2k over Z4 + uZ4 is given and their duals are determined in each case. All self-orthogonal and self-dual codes of length 2k over this ring are presented. Enumeration of codes of a particular type has been an interesting problem in coding theory. We have enumerated negacyclic codes and self-dual negacyclic of length 2k over Z4 + uZ4. This study has further been generalized to negacyclic codes arbitrary even length over Z4+uZ4. The classi cation of negacyclic codes led to some good Z4-codes via the Gray map.