2016/03/11 by Odessa D. Consorte, Consorte, Odessa D., Lilibeth D. Valdez +1
Computer Science · Engineering · Mathematics · #94B15 #Coding theory and cryptography #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Information Theory (cs.IT) #Number Theory (math.NT) #cs.IT #graph theory and CDMA systems #math.IT #math.NT #msc:94B15
paper · pdf · doi:10.48550/arxiv.1603.03520
arxiv created 2016/03/11 · openalex publication_date 2016/03/11 · arxiv updated 2016/03/14 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Cyclic and self-dual codes are important classes of codes in coding theory. Jia, Ling and Xing \citeJia as well as Kai and Zhu \citeKai proved that Euclidean self-dual cyclic codes of length n over \mathbbFq exist if and only if n is even and q=2r, 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=2ν⋅ n, where n is odd, over \mathbbF2r 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 2r-cyclotomic cosets mod n. We also express the formula for the number of cyclic self-dual codes over \mathbbF2r for each n and r in terms of the number of 2r-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 \mathbbF22 ℓ based on the existence of a nontrivial splitting (Z, X0, X1) of ℤ_n by μ-2^ℓ, where Z, X0,X1 are unions of 22 ℓ-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.