Binary codes from subset inclusion matrices

TL;DR

This paper investigates the minimum distances of binary linear codes constructed from subset inclusion matrices, providing bounds and exact values for certain parameters, and linking these codes to design theory and error correction applications.

Contribution

It introduces new bounds and exact minimum distance values for codes from subset inclusion matrices, expanding understanding of their structure and applications.

Findings

Derived bounds on minimum distances of the codes.

Exact minimum distances for t ≤ 3 and large n.

Connections established with LDPC and locally recoverable codes.

Abstract

In this paper, we study the minimum distances of binary linear codes with parity check matrices formed from subset inclusion matrices $W_{t,n,k}$, representing $t$-element subsets versus $k$-element subsets of an $n$-element set. We provide both lower and upper bounds on the minimum distances of these codes and determine the exact values for any $t\leq 3$ and sufficiently large $n$. Our study combines design and integer linear programming techniques. The codes we consider are connected to locally recoverable codes, LDPC codes and combinatorial designs.