On Euclidean and Hermitian Self-Dual Cyclic Codes over $\mathbb{F}_{2^r}$

Abstract

Cyclic and self-dual codes are important classes of codes in coding theory. Jia, Ling and Xing [5] as well as Kai and Zhu [7] proved that Euclidean self-dual cyclic codes of length n over Fq exist if and only if n is even and q = 2 r , where r is any positive integer. For n and q even, there always exists an [n, n 2 ] self-dual cyclic code with generator polynomial x n 2 + 1 called the trivial self-dual cyclic code. In this paper we prove the existence of nontrivial self-dual cyclic codes of length n = 2 ν•n, where n is odd, over F2r in terms of the existence of a nontrivial splitting (Z, X0, X1) of Z n by µ−1, where Z, X0, X1 are unions of 2 r -cyclotomic cosets mod n. We also express the formula for the number of cyclic self-dual codes over F2r for each n and r in terms of the number of 2 r -cyclotomic cosets in X0 (or in X1).We also look at Hermitian self-dual cyclic codes and show properties which are analogous to those of Euclidean self-dual cyclic codes. That is, the existence of nontrivial Hermitian self-dual codes over F 2 2ℓ based on the existence of a nontrivial splitting (Z, X0, X1) of Zn by µ −2 ℓ , where Z, X0, X1 are unions of 2 2ℓ -cyclotomic cosets mod n. We also determine the lengths at which nontrivial Hermitian self-dual cyclic codes exist and the formula for the number of Hermitian self-dual cyclic codes for each n.