GROUP RING AND ITS APPLICATION IN THE CONSTRUCTION OF SOME EXTREMAL SELF-DUAL CODES

Abstract

In this thesis, the new construction of extremal Type I and Type II self-dual codes of various lengths has been done using the group ring. Due to the numerous theoretical and practical applications of group rings and algebraic coding theory in cryptography and error correction, these topics have received much research attention. The thesis is divided into seven chapters. Chapter 1 includes relevant definitions and concepts from the literature that are pertinent to the topics employed in this thesis. The second chapter focuses on constructing extremal self-dual codes of length 16. For the first time, they are generated using the unitary units in a group ring with the Quaternion group. Various code modification techniques are being applied in the correct order to self-dual codes, which improves the rates (ratio of information symbol to code length) and error-handling capacity of the code. Chapter three focuses on a new construction for self-dual codes that uses the con cept of double-bordered construction, group rings, and reverse circulant matrices. Us ing groups of orders 2, 3, 4, and 5, and by applying the construction over the binary field F2 and the ring F2 + uF2, an extremal binary self-dual codes of various lengths: 12, 16, 20, 24, 32, 40, and 48 are obtained. The significance of this new construction is the construction of the unique Extended Binary Golay Code [24, 12, 8], and the unique Ex tended Quadratic Residue [48, 24, 12] Type II linear block code. Moreover, the existing relationship between units and non-units with the self-dual codes presented in (23) is also strengthened by limiting the conditions given in the corollaries of (23). Additionally, a relationship between idempotent and self-dual codes is also established. In chapter four the concept of n r -th borders around the matrix is introduced. Here n and r are the natural numbers such that r divides n. We have shown that this construction is efficacious for any groups of order r (where r is a natural number such that r divides n), over the Frobenius ring Rk . We discover extremal binary self-dual codes of lengths 32, 40, the well-known Extended Binary Golay Code, i.e., [24, 12, 8], and Extended Quadratic Residue Code, i.e., [48, 24, 12] by two different ways. xi xii Abstract In chapter five, we introduce the double-bordered construction of self-dual codes whose generator matrix is of the form M = [In|A] where A is a block matrix consisting of blocks that come from group rings and the elements in the first row cannot completely determine the block matrix A. We demonstrate that this construction is feasible for a group of order 2n where n is a natural number, over the Frobenius ring Rk . We show the significance of this new construction by constructing several extremal self-dual codes of lengths 20, 40, 32, and 64 over the field F2 and the ring F2 + uF2. Chapter six focuses on the new technique for the construction of self-dual codes. Double borders are introduced around a new altered form of a four-circulant matrix. Us ing this new construction over the field F2 and the ring F2 + uF2 and groups of orders 2, 3, 4, 5, 7, and 9, we generate extremal binary self-dual codes of the following lengths: 12, 20, 24, 32, 40, 48, 64, and 80. In chapter seven we introduce a new class of ring, which is the ∗-version of the semiclean ring, i.e., the ∗-semiclean ring. A ∗-ring is ∗-semiclean if each element is the sum of a ∗-periodic element and a unit. Many properties of ∗-semiclean rings are discussed. It is proved that if p ∈ P(R) such that pRp and (1 − p)R(1 − p) are ∗-semiclean rings, then R is also a ∗-semiclean ring. As a result, the matrix ring Mn(R) over a ∗- semiclean ring is ∗-semiclean. A characterization that when the group rings RCr and RG are ∗-semiclean is done, where R is a finite commutative local ring, Cr is a cyclic group of order r, and G is a locally finite abelian group. We have also found sufficient conditions when the group rings RC3, RC4, RQ8, and RQ2n are ∗-semiclean, where R is a commutative local ring. We have also demonstrated that the group ring Z2D6 is a ∗- semiclean ring (which is not a ∗-clean ring). We have characterized the ∗-semicleanness of FqG in terms of LCD and self-orthogonal abelian codes under the classic involution, where Fq is a finite field with q elements and G is a finite abelian group.