Moment-based linear programming bounds for locally recoverable codes
TL;DR
New Delsarte-type linear programming bounds for $q$-ary $(r,\delta)$-locally recoverable codes (LRCs) are derived with three attributes: the variable set is comparable in size to that of the classical Delsarte LP, and the convex-hull LP yields improvements not captured by the previous LP and often sharpens the shortening and generalized Singleton bounds.
Abstract
In this paper we derive new Delsarte-type linear programming bounds for $q$-ary $(r,\delta)$-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 \(\delta-2\), 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 $\delta \ge 3$, give a further strengthening; and third, it applies to general $(r,\delta)$-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.