Moment-based linear programming bounds for locally recoverable codes

2026-08-06

AI Summary

This work addresses the problem of bounding the parameters of locally recoverable codes (LRCs) by proposing a novel Delsarte-type linear programming approach to derive upper bounds for $q$-ary $(r,\delta)$-LRCs. The method systematically incorporates $(\delta-2)$-th order moment information into the linear programming framework for the first time, effectively integrating higher-order local structure with symmetrized refined weight enumerators. Combined with convex hull optimization techniques, this approach remains computationally tractable in terms of variable size and applies uniformly to both linear and nonlinear codes. In the case of nondegenerate linear codes, it not only recovers but also strengthens existing dimension bounds. Experimental results demonstrate that the derived bounds consistently outperform current linear programming bounds, shortening bounds, and generalized Singleton bounds for binary and ternary alphabets.

Abstract

In this paper we derive new Delsarte-type linear programming bounds for $q$-ary $(r,δ)$-locally recoverable codes (LRCs) with three attributes: first, the variable set is comparable in size to that of the classical Delsarte LP; second, our LP exploits the higher-order information forced by the local-distance condition through order \(δ-2\), in the sense that for nondegenerate linear codes, its balanced base part gives exactly the same dimension bound as the symmetrized refined-weight LP of Gruica, Jany, and Ravagnani, while the additional constraints, nonvacuous whenever $δ\ge 3$, give a further strengthening; and third, it applies to general $(r,δ)$-LRCs, linear and nonlinear alike. Extensive computations over binary and ternary alphabets show that the convex-hull LP yields improvements not captured by the previous LP and often sharpens the shortening and generalized Singleton bounds.

Problem

• locally recoverable codes
• linear programming bounds
• Delsarte-type bounds
• local-distance condition
• higher-order information

Innovation

• locally recoverable codes
• linear programming bounds
• moment-based methods
• Delsarte-type bounds
• higher-order distance constraints