Binary codes from subset inclusion matrices

2024/08/22 by Marin, Alexey D., I. Yu. Mogilnykh, Mogilnykh, Ivan Yu.
Computer Science · Engineering · #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2408.12154

openalex publication_date 2024/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≤ 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.

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