Upper bound on list-decoding radius of binary codes

TL;DR

This paper establishes an asymptotic upper bound on the rate of binary codes with list-decoding constraints, improving previous bounds for certain rates by combining linear programming and Ramsey theory techniques.

Contribution

It introduces a new upper bound on list-decoding radius for binary codes with odd list sizes, enhancing prior results using a novel combination of methods.

Findings

Improved upper bound on code rate for odd list sizes

Bound is tighter than Blinovsky's for certain rates

Shows zero slope of rate-radius tradeoff at zero rate for all odd L

Abstract

Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most $L$. For odd $L\ge 3$ an asymptotic upper bound on the rate of any such packing is proven. Resulting bound improves the best known bound (due to Blinovsky'1986) for rates below a certain threshold. Method is a superposition of the linear-programming idea of Ashikhmin, Barg and Litsyn (that was used previously to improve the estimates of Blinovsky for $L=2$) and a Ramsey-theoretic technique of Blinovsky. As an application it is shown that for all odd $L$ the slope of the rate-radius tradeoff is zero at zero rate.