TL;DR
This paper introduces a new hierarchy of linear programs that converges to the true size of optimal linear codes, offering a potentially more analyzable approach to bounding code parameters than previous methods.
Contribution
It presents a complete LP hierarchy generalizing Delsarte's LPs, converging to the true code size, and applicable to translation schemes, advancing coding theory bounds.
Findings
Hierarchy converges to the true code size at level O(n^2)
Generalizes Delsarte LPs using higher-order Krawtchouk polynomials
Potentially more amenable to theoretical analysis
Abstract
A longstanding open problem in coding theory is to determine the best (asymptotic) rate of binary codes with minimum constant (relative) distance . An existential lower bound was given by Gilbert and Varshamov in the 1950s. On the impossibility side, in the 1970s McEliece, Rodemich, Rumsey and Welch (MRRW) proved an upper bound by analyzing Delsarte's linear programs. To date these results remain the best known lower and upper bounds on with no improvement even for the important class of linear codes. Asymptotically, these bounds differ by an exponential factor in the blocklength. In this work, we introduce a new hierarchy of linear programs (LPs) that converges to the true size of an optimum linear binary code (in fact, over any finite field) of a given blocklength and distance . This hierarchy has several notable…
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Videos
A Complete Linear Programming Hierarchy for Linear Codes· youtube
