On self-dual double circulant codes

Abstract

Self-dual double circulant codes of odd dimension are shown to be dihedral in even characteristic and consta-dihedral in odd characteristic. Exact counting formulae are derived for them and used to show they contain families of codes with relative distance satisfying a modified Gilbert-Varshamov bound.

On self-dual double negacirculant codes

“…The existence of infinitely many such n ′ s requires the truth of Artin's conjecture [14]. Thus the present paper is more explicit and more elementary than [1].…”

“…By an old result of [18], a DC self-dual code does not exist over fields like F 3 where −1 is not a square. This paper can thus be considered as a companion paper of [1]. The main difference between the two papers is that a factorization of x n + 1 into two irreducible polynomials is easier and more elementary to find than a factorization of x n − 1 into two irreducible polynomials which requires n to be a prime for which 2 is primitive.…”

On Self-Dual Four Circulant Codes

“…In the present paper, we study index 4 quasi-cyclic codes of parameters [4n, 2n] inspired by [1,3,8]. When x n − 1 has only two irreducible factors, we derive an enumeration formula of the self-dual four circulant codes over F q based on that decomposition, and derive an asymptotic lower bound on the minimum distance of these four circulant codes.…”

Asymptotically Good Additive Cyclic Codes Exist

“…Thus, binary quasi-cyclic codes of index 2 were shown to be asymptotically [13]. This was extended to any finite field, and refined to self-dual codes in [2]. The result was further extended to any index in [1].…”