TY - JOUR

T1 - Binary Codes From Subset Inclusion Matrices

AU - Marin, Alexey D.

AU - Mogilnykh, Ivan Yu

N1 - The authors would like to express their gratitude to Alexey Frolov for a talk on locally recoverable codes at the online seminar“Coding theory,” Vladimir Potapov for talks on unitrades at the seminar “2024‐ary quasigroups and related topics.” These contri-butions significantly directed their focus toward the current research. The authors are profoundly thankful to Evgeny Vdovin forproviding the licensed MAGMA software, which was indispensable for this study. They also thank the anonymous referees forvaluable remarks and suggestions. This study was performed according to the Government research assignment for the SobolevInstitute of Mathematics, Siberian Branch of the Russian Academy of Sciences, project FWNF‐2022‐0017.

PY - 2025

Y1 - 2025

N2 - In this paper, we study the minimum distances of binary linear codes with parity check matrices formed from subset inclusion matrices (Formula presented.), representing (Formula presented.) -element subsets versus (Formula presented.) -element subsets of an (Formula presented.) -element set. We provide both lower and upper bounds on the minimum distances of these codes and determine the exact values for any (Formula presented.) and sufficiently large (Formula presented.). Our study combines design and integer linear programming techniques. The codes we consider are connected to LDPC codes and combinatorial designs. Furthermore, we construct quasi-cyclic LDPC codes from inclusion matrices that exhibit performance comparable to or slightly better than MacKay-type codes when evaluated using bit flipping and min-sum algorithms.

AB - In this paper, we study the minimum distances of binary linear codes with parity check matrices formed from subset inclusion matrices (Formula presented.), representing (Formula presented.) -element subsets versus (Formula presented.) -element subsets of an (Formula presented.) -element set. We provide both lower and upper bounds on the minimum distances of these codes and determine the exact values for any (Formula presented.) and sufficiently large (Formula presented.). Our study combines design and integer linear programming techniques. The codes we consider are connected to LDPC codes and combinatorial designs. Furthermore, we construct quasi-cyclic LDPC codes from inclusion matrices that exhibit performance comparable to or slightly better than MacKay-type codes when evaluated using bit flipping and min-sum algorithms.

KW - binary design

KW - combinatorial design

KW - inclusion matrix

KW - minimum distance

UR - https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=105018685141&origin=inward

UR - https://www.mendeley.com/catalogue/c05805df-4255-369e-88f3-0e134e9c3012/

U2 - 10.1002/jcd.22012

DO - 10.1002/jcd.22012

M3 - Article

JO - Journal of Combinatorial Designs

JF - Journal of Combinatorial Designs

SN - 1063-8539

ER -