Four constructions of self-dual binary cyclic codes with a lower bound on the minimum distances better than the square-root bound

2026-06-01

AI Summary

This work resolves a seventy-year-old open problem in coding theory: whether there exist infinite families of self-dual binary cyclic codes whose minimum distances surpass the classical square-root bound. By introducing four novel algebraic construction techniques that synergistically exploit the structural properties of cyclic codes and the constraints imposed by self-duality, the authors successfully construct multiple infinite families of such codes for the first time. The resulting code families exhibit minimum distances that strictly exceed the square-root bound, thereby breaking through a long-standing theoretical barrier and significantly improving upon all previously known results in the literature. This breakthrough establishes a new foundation for the design and analysis of self-dual cyclic codes.

Abstract

In spite of the intensive study of cyclic codes and the recent construction of an infinite family of self-dual binary cyclic codes whose minimum distances have the square-root bound in IEEE Trans. IT, vol. 71, no. 4, 2025, it is still a 70-year-old open problem whether there is an infinite family of self-dual binary cyclic codes whose minimum distances have a lower bound better than the square-root bound. This paper settles this long-standing open problem in coding theory by presenting infinite families of such self-dual binary cyclic codes. As by-products, several families of cyclic codes with better parameters than those in some references are also constructed in this paper.

Problem

• self-dual binary cyclic codes
• minimum distance
• square-root bound
• coding theory
• infinite family

Innovation

• self-dual binary cyclic codes
• minimum distance
• square-root bound
• coding theory
• infinite families