A STUDY ON SKEW CODES AND QUANTUM 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 channels. Since its inception, it has grown in to a large area, intersecting several disciplines and using several sophisticated mathematical techniques. The branch of coding theory that mainly uses algebraic tools is known as Algebraic Coding Theory. Initially, algebraic codes were constructed as vector spaces over finite fields. However, later on many rings have also been considered in place of fields, and codes were studied as modules over finite rings. A recent addition to coding theory literature is skew codes, in which algebraic codes are constructed using skew polynomial rings. Several results have been obtained on these codes, and many new good codes have been obtained in this setting. This thesis deals with some families of codes in the setting of skew polynomial rings over some extensions of Z4 and Fq, where Z4 is the ring of integers modulo 4 and Fq is a finite field. These are skew-cyclic codes, skew-constacyclic codes, 2D skew-cyclic codes etc. In addition, quantum codes over F4 + uF4 have also been studied. In this context, we have defined a new class of skew-cyclic codes over the mixed alphabet F3(F3 + vF3), v2 = v. We call these codes F3(F3 + vF3)-skew cyclic codes, and they can be seen as a generalization of double cyclic codes [25] and Z2(Z2+uZ2)- linear cyclic codes [6]. We have obtained a structure of skew-cyclic codes over F3+vF3 by defining a division algorithm on (F3+vF3)[x, ]. Using this structure, we have obtained the structures of F3(F3 +vF3)-skew cyclic codes and their generating i ii sets. The duals of these codes have also been studied. Also, we have studied a class of skew-cyclic codes over Fp + wFp,w2 = 1, wherein the generating sets of these codes have been obtained. The extensions of Z4 such as Z4 + uZ4 have attracted the attention of a lot of researchers in last few years. Some studies have shown that the codes over these rings are promising and can produce codes with better parameters. However, there has been a relatively little study on skew codes over these types of rings. We study a class of skew-constacyclic codes over the ring Z4 + uZ4, u2 = 0. By defining an automorphism on Z4 + uZ4, we study these codes as left (Z4 + uZ4)[x, ]- submodules of (Z4+uZ4)[x, ] hxn− i , where = 1 + 2u, a unit in Z4 + uZ4. A necessary and sufficient condition for a skew-constacyclic code over Z4 +uZ4 to be principally generated has been obtained. Duals of these codes have also been studied and these codes have been further generalized to double skew-constacylic codes. By finding the Gray images of these codes some new good Z4-linear codes having parameters (6, 4422, 2L), (18, 4421, 10L), (18, 4422, 7L) and (18, 4424, 7L) have been obtained. Moreover, we have reported these codes to the database of Z4-codes [8]. A class of skew-cyclic codes over the ring GR(4, 2)+vGR(4, 2), v2 = v, has also been studied. We have also studied skew codes in the more general setting of a skew-polynomial ring with automorphism and derivation. In this context, we have studied a class of skew-cyclic codes over Z4 + wZ4,w2 = 1, with derivation. We denote these codes by -cyclic codes. These codes are studied as left (Z4 + wZ4)[x, , ]-submodules of (Z4+wZ4)[x, , ] hxn−1i , where is an automorphism of Z4 + wZ4 and a derivation on Z4 + wZ4. Using a Gray map, some good linear codes over Z4, via residue codes of these codes, have been obtained. A generator matrix of the dual code of a free -cyclic code of even length over Z4 + wZ4 has been obtained. These codes are further generalized to double skew-cyclic codes with derivation. The classification of these codes also led to some new good Z4-codes. There is another generalization of cyclic codes, known as 2D cyclic codes. Recently, Li & Li [65] have studied 2D skew-cyclic codes over a finite field Fq. We iii generalize the study of 2D skew-cyclic codes over Fq to 2D skew-cyclic codes over Fq+wFq,w2 = 1. The structure of these codes has been obtained by defining a division algorithm on the bivariate polynomial ring (Fq + wFq)[x, y, 1, 2], where 1, 2 are two commuting automorphisms of Fq + wFq. These codes have been studied as left (Fq + wFq)[x, y, 1, 2]-submodules of (Fq+wFq)[x,y, 1, 2] hxl−1, ym−1i . A brief description of the duals of these codes has also been given. A decomposition of these codes has been presented, via which a generating set of a 2D skew-cyclic code over Fq + wFq is determined using generating sets of its component 2D skew-cyclic codes over Fq. The relationship between quantum information and classical information has become a subject of much study in recent years. The construction of quantum codes using classical linear codes was given by Calderbank et al. [31]. Motivated by the recent progress in this field, we have studied quantum codes over F4+uF4, u2 = 0. In our study, we use the structure of cyclic codes of arbitrary length over F4+uF4 to find out the conditions for these codes to contain their duals. By the CSS construction and a Gray map, the parameters of the corresponding quantum codes over F4 have been obtained. Also, using augmentation, we enlarge a code with dual containing property to a new code having the same property, and we have got some good quantum codes over F4 using this technique. A table showing some good quantum codes that we have obtained over F4 is also given.