2024/11/21 by Minjia Shi, Shitao Li, Shi, Minjia +5
Computer Science · #Coding theory and cryptography #Cooperative Communication and Network Coding #Error Correcting Code Techniques #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2411.14087
openalex publication_date 2024/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For an integer s≥ 1, let Cs(q0) be the generalized Zetterberg code of length q0s+1 over the finite field \Fq0 of odd characteristic. Recently, Shi, Helleseth, and Özbudak (IEEE Trans. Inf. Theory 69(11): 7025-7048, 2023) determined the covering radius of Cs(q0) for q0s \not ≡ 7 \pmod8, and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for q0s ≡ 7 \pmod8, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length (q0s+1)/(2), and show the same results hold for them. As a result, we obtain some quasi-perfect codes.