Cyclic codes over \mathbbF_2^m[u]/⟨ u^k⟩ of oddly even length

2015/11/17 by Yonglin Cao, Yuan Cao, Cao, Yonglin +3
Computer Science · Engineering · #Coding theory and cryptography #Cellular Automata and Applications #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1511.05413

Abstract

Let \mathbbF_2^m be a finite field of characteristic 2 and R=\mathbbF_2^m[u]/⟨ u^k⟩=\mathbbF_2^m +u\mathbbF_2^m+…+u^k-1\mathbbF_2^m (u^k=0) where k∈ ℤ⁺ satisfies k≥ 2. For any odd positive integer n, it is known that cyclic codes over R of length 2n are identified with ideals of the ring R[x]/⟨ x²ⁿ-1⟩. In this paper, an explicit representation for each cyclic code over R of length 2n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length 2n is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over R of length 2n are investigated. (AAECC-1522)

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