Multi-twisted additive self-orthogonal and ACD codes are asymptotically good

Abstract

Multi-twisted (MT) additive codes over finite fields constitute an important class of additive codes and are generalizations of cyclic and constacyclic additive codes. In this paper, we employ probabilistic methods and results from groups and geometry to study the asymptotic behavior of the rates and relative Hamming distances of two special subclasses of MT additive codes over finite fields, viz. MT additive self-orthogonal codes and MT additive codes with complementary duals (MT ACD codes) with respect to ordinary, Hermitian, and ⁎ trace bilinear forms. More precisely, let p be an odd prime, q be a prime power coprime to p, v be the multiplicative order of q modulo p, and let η be the largest positive integer such that pη|(qv−1). Here we establish the existence of asymptotically good infinite sequences of MT additive self-orthogonal and ACD codes of length pαℓ and block length pα over Fqt with pα→∞, relative Hamming distance at least δ and rates vpηℓt and 2vpηℓt with respect to the aforementioned trace bilinear forms, where ℓ≥1, t≥2 are integers, Fqt is the finite field of order qt and δ is a positive real number satisfying hqt(δ)<12−12ℓt, (here hqt is the qt-ary entropy function). This shows that MT additive self-orthogonal and ACD codes over finite fields are asymptotically good. As a special case, we deduce that constacyclic additive self-orthogonal and ACD codes over finite fields are asymptotically good.