On self-dual negacirculant codes of index two and four

• Published in: Designs, codes, and cryptography Vol. 86; no. 11; pp. 2485 - 2494
• Main Authors: Shi, Minjia, Qian, Liqin, Solé, Patrick
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2018; Springer Nature B.V; Springer Verlag
• Subjects: Circuits; Coding and Information Theory; Computer Science; Cryptology; Data Structures and Information Theory; Discrete Mathematics in Computer Science; Enumeration; Information and Communication; Information Theory; Number theory; 94 B25; Self-dual codes; 94 B15; 05 E30; Negacirculant codes; Self-reciprocal polynomials; Artin primitive root conjecture
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-017-0455-0

We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring R = F q [ x ] / ⟨ x n + 1 ⟩ , when they are self-dual and of length 2 n or 4 n . In particular, in order for the decomposition to be amenable to analysis, we study factorizations of x n + 1 over F q , with n twice an odd prime, containing only three irreducible factors, all self-reciprocal. We give arithmetic conditions bearing on n and q for this to happen. Given a fixed q ,  we show these conditions are met for infinitely many n ’s, provided a refinement of Artin primitive root conjecture holds. This number theory conjecture is known to hold under generalized Riemann hypothesis (GRH). We derive a modified Varshamov–Gilbert bound on the relative distance of the codes considered, building on exact enumeration results for given n and q .