On some special classes of additive codes over finite fields

Abstract

In this thesis, we define and study a new class of additive codes over finite fields, viz. multi-twisted (MT) additive codes, which is a generalization of constacyclic additive codes and an extension of MT (linear) codes introduced by Aydin and Haliloviand#263; [5]. We study their algebraic structures by writing a canonical form decomposition of these codes using the Chinese Remainder Theorem and provide an enumeration formula for these codes. With the help of their canonical form decomposition, we also provide a trace description for all MT additive codes over finite fields. We further apply probabilistic methods to study the asymptotic properties of the rates and relative Hamming distances of a special subclass of 1-generator MT additive codes. We show that there exists an asymptotically good infinite sequence of MT additive codes of length pand#8629;` and block length pand#8629; ! 1 over Fqt with rate v pand#8984;`t and relative Hamming distance at least , where ` 1 and t 2 are integers, q is a prime power, Fqt is the finite field of order qt , p is an odd prime satisfying gcd(p, q)=1, v = ordp(q) is the multiplicative order of q modulo p, and#8984; is the largest positive integer such that pand#8984; | (qv 1) and is a positive real number satisfying hqt () lt 1 1 `t, (here hqt (·) denotes the qt -ary entropy function). This shows that the family of MT additive codes over finite fields is asymptotically good. As special cases, we deduce that the families of constacyclic and cyclic additive codes over finite fields are asymptotically good. By placing ordinary, Hermitian and and#8676; trace bilinear forms, we study the dual codes of MT additive codes over finite fields and derive necessary and sufficient conditions under which an MT additive code is (i) self-orthogonal, (ii) self-dual and (iii) an additive code with complementary dual (or an ACD code). We also derive a necessary and sufficient condition for the existence of a self-dual MT additive code over a finite field and provide enumeration formulae for all self-orthogonal, self-dual and ACD MT additive codes ov