New Constructions of Binary Cyclic Codes with Both Relatively Large Minimum Distance and Dual Distance

[2504.11010] New Constructions of Binary Cyclic Codes with Both Relatively Large Minimum Distance and Dual Distance Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Information Theory arXiv:2504.11010 (cs) [Submitted on 15 Apr 2025 ( v1 ), last revised 11 Apr 2026 (this version, v2)] Title: New Constructions of Binary Cyclic Codes with Both Relatively Large Minimum Distance and Dual Distance Authors: Lingqi Zheng , Weijun Fang , Rongxing Qiu View a PDF of the paper titled New Constructions of Binary Cyclic Codes with Both Relatively Large Minimum Distance and Dual Distance, by Lingqi Zheng and 2 other authors View PDF HTML (experimental) Abstract: Binary cyclic codes are worth studying due to their applications and theoretical importance. It is an important problem to construct an infinite family of cyclic codes with large minimum distance $d$ and dual distance $d^{\perp}$. In recent years, much research has been devoted to improving the lower bound on $d$, some of which have exceeded the square-root bound. The constructions presented recently seem to indicate that when the minimum distance increases, the minimum distance of its dual code decreases. In this paper, we focus on the new constructions of binary cyclic codes with length $n=2^m-1$, dimension near $n/2$ and both relatively large minimum distance and dual distance. When $m$ is even, we construct a family of binary cyclic codes with parameters $[2^m-1,2^{m-1}\pm1,d]$, where $d\ge 2^{m/2}-1$ and $d^\perp\ge2^{m/2}$. Both the minimum distance and the dual distance are significantly better than the previous results. When $m$ is the product of two distinct primes, we construct some cyclic codes with dimensions $k=(n+1)/2$ and $d>\frac{n}{\log_2n},$ where the lower bound on the minimum distance is much larger than the square-root bound. When $m$ is odd, we present two families of binary $[2^m-1,2^{m-1},d]$ cyclic codes with $d\ge2^{(m+1)/2}-1$, $d^\perp\ge2^{(m+1)/2}$ and $d\ge2^{(m+3)/2}-15$, $d^\perp\ge2^{(m-1)/2}$ respectively, which leads that $d\cdot d^\perp$ can reach $2n$ asymptotically. To the best of our knowledge, for the binary cyclic codes with length $n=2^m-1$ and dimension $k=(n\pm1)/2$, except for the punctured binary Reed-Muller codes, there is no other construction of binary cyclic codes that reaches this bound. Comments: Accepted for publication in IEEE Transactions on Information Theory Subjects: Information Theory (cs.IT) Cite as: arXiv:2504.11010 [cs.IT] (or arXiv:2504.11010v2 [cs.IT] for this version) https://doi.org/10.48550/arXiv.2504.11010 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Weijun Fang [ view email ] [v1] Tue, 15 Apr 2025 09:29:41 UTC (26 KB) [v2] Sat, 11 Apr 2026 06:12:34 UTC (25 KB) Full-text links: Access Paper: View a PDF of the paper titled New Constructions of Binary Cyclic Codes with Both Relatively Large Minimum Distance and Dual Distance, by Lingqi Zheng and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.IT < prev | next > new | recent | 2025-04 Change to browse by: cs math math.IT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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