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 Halilović [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 p↵` and block length p↵ ! 1 over Fqt with rate v p⌘`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, ⌘ is the largest positive integer such that p⌘ | (qv 1) and is a positive real number satisfying hqt () < 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 ⇤ 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 over finite fields with respect to the aforementioned trace bilinear forms. We further employ probabilistic methods and results from groups and geometry to study the asymptotic behavior of the rates and relative Hamming distances of self-orthogonal, self-dual and ACD MT additive codes over finite fields with respect to the aforementioned trace bilinear forms. We establish the existence of asymptotically good infinite sequences of self-orthogonal and ACD MT additive codes of length p↵` and block length p↵ ! 1 over Fqt with relative Hamming distance at least and rates v p⌘`t and 2v p⌘`t with respect to the aforementioned trace bilinear forms, where is a positive real number satisfying hqt () < 1 2 1 2`t, p is an odd prime coprime to q, v = ordp(q) is the multiplicative order of q modulo p, and ⌘ is the largest positive integer satisfying p⌘ | (qv 1). This shows that self-orthogonal and ACD MT additive codes over finite fields are asymptotically good. As special cases, we deduce that self-orthogonal and ACD constacyclic additive codes over finite fields are asymptotically good. We also establish the existence of asymptotically good infinite sequences of self-dual MT additive codes of length p↵` and block length p↵ ! 1 over Fqt with relative Hamming distance at least and rate 1 2 with respect to the aforementioned trace bilinear forms, where is a positive real number satisfying hqt () < 1 2`t, p is an odd prime coprime to q such that v = ordp(q) is even. As special cases, we deduce that self-dual cyclic and negacyclic additive codes over finite fields are asymptotically good. We also define and study a special class of self-dual MT additive codes over Fqt with respect to the ordinary trace bilinear form, viz. doubly-even self-dual MT additive codes and characterize a special class of these codes in terms of their constituents with the help of their trace description, where q is an even prime power. With the help of this characterization and using probabilistic methods and results from groups and geometry, we further study the asymptotic behaviour of their relative Hamming distances and show that doublyeven self-dual MT additive codes over F2t are asymptotically good. As a special case, we also deduce that doubly-even self-dual cyclic additive codes over F2t are asymptotically good.We next define and study a new class of additive codes over finite fields, viz. additive quasi-Abelian (QA) codes, which is a generalization of a special class of MT additive codes over finite fields and an extension of linear QA codes over finite fields. We study the algebraic structures of these codes and their dual codes with respect to ordinary, Hermitian and ⇤ trace bilinear forms. We further express these codes as direct sums of additive codes over finite fields and derive necessary and sufficient conditions under which an additive QA code is (i) self-orthogonal, (ii) self-dual and (iii) ACD. We also derive necessary and sufficient conditions for the existence of a self-dual additive QA code over finite fields. Besides this, we obtain explicit enumeration formulae for all self-orthogonal, self-dual and ACD additive QA codes over finite fields. We also list several MDS and almost MDS codes belonging to the family of additive QA codes over finite fields, which shows that additive QA codes over finite fields is a promising class of codes to find codes with good and optimal parameters.