Linear Programming Bounds for Almost-Balanced Binary Codes

TL;DR

This paper analyzes linear programming bounds for almost-balanced binary codes, providing optimal solutions in certain cases and exploring limitations of asymptotic bounds, thereby advancing understanding of code size versus distance trade-offs.

Contribution

It offers an optimal solution to Delsarte's LP for almost-balanced codes with large distance and examines the limitations of asymptotic LP bounds in this context.

Findings

Optimal LP solution matches Grey-Rankin bound for self-complementary codes.

Asymptotic LP bounds are at least the average of MRRW and Gilbert-Varshamov bounds in the almost-balanced case.

Limitations of asymptotic bounds persist for the almost-balanced code scenario.

Abstract

We revisit the linear programming bounds for the size vs. distance trade-off for binary codes, focusing on the bounds for the almost-balanced case, when all pairwise distances are between $d$ and $n-d$, where $d$ is the code distance and $n$ is the block length. We give an optimal solution to Delsarte's LP for the almost-balanced case with large distance $d \geq (n - \sqrt{n})/2 + 1$, which shows that the optimal value of the LP coincides with the Grey-Rankin bound for self-complementary codes. We also show that a limitation of the asymptotic LP bound shown by Samorodnitsky, namely that it is at least the average of the first MRRW upper bound and Gilbert-Varshamov bound, continues to hold for the almost-balanced case.