On Euclidean and Hermitian self-dual cyclic codes over GF(2r) - Consorte, Odessa D.; Valdez, Lilibeth D. (2017)
Jia et al. (2011) and Jitman et al. (2014) characterized Euclidean and Hermitian self-dual cyclic codes, respectively, by considering reciprocal and conjugate-reciprocal factors of the generator polynomial of these codes. In this paper, we give an alternative approach to this study by using splittings and cyclomatic cosets. We prove the existence of nontrivial Euclidean self-dual cyclic codes of length n = 2v · ñ, where ñ is odd, over GF (2r) in terms of the existence of a nontrivial splitting (Z, X0, X1) of Zñ by µ-1, where Z, X0, X1 are unions of 2r-cyclomatic cosets mod ñ. We express the formula for the number of cyclic self-dual codes over GF(2r) for each n and r in terms of the number of 2r-cyclomatic cosets in X0 (or in X1). In addition, we look at Hermitian self-dual cyclic codes. Nontrivial Hermitian self-dual codes over GF(22e) exist based on the existence of a nontrivial splitting (Z, X0, X1) of Zñ by µ-2e, where (Z, X0, X1) are unions of 22e-cyclomatic cosets mod ñ. From this splitting, we give a formula for the number of Hermitian self-dual cyclic codes for each n. Furthermore, we give an arithmetic condition on the length n such that nontrivial Hermitian self-dual cyclic codes exist. (Author's abstract)