Cyclic codes over ℤ₄[u]/⟨ u^k⟩ of odd length

2016/06/15 by Yuan Cao, Qingguo Li, Yuan, Cao +1
Computer Science · Engineering · #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1606.04601

openalex publication_date 2016/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R=ℤ₄[u]/⟨ u^k⟩=ℤ₄+uℤ₄+…+u^k-1ℤ₄ (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 n are identified with ideals of the ring R[x]/⟨ xⁿ-1⟩. In this paper, an explicit representation for each cyclic code over R of length n 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 n is obtained. Precisely, the dual code of each cyclic code and self-dual cyclic codes over R of length n are investigated. When k=4, some optimal quasi-cyclic codes over ℤ₄ of length 28 and index 4 are obtained from cyclic codes over R=ℤ₄ [u]/⟨ u⁴⟩.

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