Exact Completeness of LP Hierarchies for Linear Codes

TL;DR

This paper proves that two LP hierarchies exactly determine the maximum size of binary linear codes at a certain level, providing new insights into their structural power and integrality properties.

Contribution

The authors demonstrate that both LP hierarchies recover the exact maximum code size at level n and establish the integrality of the Loyfer-Linial polytope at this level.

Findings

Both hierarchies exactly compute $A_2^{ ext{Lin}}(n,d)$ at level n.

The Loyfer-Linial polytope is integral at this level.

Hierarchies are sufficiently powerful to achieve exact results despite being less general than Sum-of-Squares.

Abstract

Determining the maximum size $A_2(n,d)$ of a binary code of blocklength $n$ and distance $d$ remains an elusive open question even when restricted to the important class of linear codes. Recently, two linear programming hierarchies extending Delsarte's LP were independently proposed to upper bound $A_2^{\text{Lin}}(n,d)$ (the analogue of $A_2(n,d)$ for linear codes). One of these hierarchies, by the authors, was shown to be approximately complete in the sense that the hierarchy converges to $A_2^{\text{Lin}}(n,d)$ as the level grows beyond $n^2$. Despite some structural similarities, not even approximate completeness was known for the other hierarchy by Loyfer and Linial. In this work, we prove that both hierarchies recover the exact value of $A_2^{\text{Lin}}(n,d)$ at level $n$. We also prove that at this level the polytope of Loyfer and Linial is integral.Even though these hierarchies seem less powerful than general hierarchies such as Sum-of-Squares, we show that they have enough structure to yield exact completeness via pseudoprobabilities.