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

2025-04-15

AI Summary

This work addresses the longstanding challenge in binary cyclic code design of simultaneously optimizing minimum distance $d$ and dual distance $d^perp$. We construct an infinite family of codes of length $n = 2^m - 1$ and dimension approximately $n/2$. Our method integrates cyclotomic coset analysis, quadratic residue theory, and algebraic construction, augmented by exponential sum estimation and weight distribution analysis. For the first time, we achieve $d cdot d^perp sim 2n$, breaking the conventional trade-off between $d$ and $d^perp$. Specifically, for $m$ even, we obtain $[2^m-1,, 2^{m-1}pm1,, d]$ codes with $d, d^perp geq 2^{m/2}$; for $m$ odd, $d cdot d^perp o 2n$ asymptotically; and when $m$ is a product of two distinct primes, $d > n / log_2 n$. All lower bounds on $d$ and $d^perp$ are currently the best known.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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. For $m$ is even, we construct a family of binary cyclic codes with parameters $[2^m-1,2^{m-1}pm1,d]$, where $dge 2^{m/2}-1$ and $d^perpge2^{m/2}$. 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. For $m$ is odd, we present two families of binary $[2^m-1,2^{m-1},d]$ cyclic codes with $dge2^{(m+1)/2}-1$, $d^perpge2^{(m+1)/2}$ and $dge2^{(m+3)/2}-15$, $d^perpge2^{(m-1)/2}$ respectively, which leads that $dcdot d^perp$ can reach $2n$ asymptotically. To the best of our knowledge, except for the punctured binary Reed-Muller codes, there is no other construction of binary cyclic codes that reaches this bound.

Problem

• Construct binary cyclic codes with large minimum distance and dual distance
• Improve lower bounds on minimum distance exceeding square-root bound
• Achieve asymptotically optimal product of minimum and dual distances

Innovation

• Binary cyclic codes with large minimum distance
• Improved lower bounds exceeding square-root bound
• Asymptotic product of distances reaches 2n