On self-duality and hulls of cyclic codes over \frac\mathbbF_2^m[u]⟨ u^k⟩ with oddly even length

2019/10/05 by Yonglin Cao, Yuan Cao, Cao, Yonglin +3
Computer Science · Engineering · #Cellular Automata and Applications #Coding theory and cryptography #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1910.02237

openalex publication_date 2019/10/05 · openalex created_date 2019/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbF2m be a finite field of 2m elements, and R=\mathbbF2m[u]/⟨ uk⟩=\mathbbF2m+u\mathbbF2m+…+uk-1\mathbbF2m (uk=0) where k is an integer satisfying k≥ 2. For any odd positive integer n, an explicit representation for every self-dual cyclic code over R of length 2n and a mass formula to count the number of these codes are given first. Then a generator matrix is provided for the self-dual and 2-quasi-cyclic code of length 4n over \mathbbF2m derived by every self-dual cyclic code of length 2n over \mathbbF2m+u\mathbbF2m and a Gray map from \mathbbF2m+u\mathbbF2m onto \mathbbF2m2. Finally, the hull of each cyclic code with length 2n over \mathbbF2m+u\mathbbF2m is determined and all distinct self-orthogonal cyclic codes of length 2n over \mathbbF2m+u\mathbbF2m are listed.

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