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Binary codes from subset inclusion matrices

Alexey D. Marin, Ivan Yu. Mogilnykh

TL;DR

This work determines the minimum distance $d_{t,n,k}$ for binary codes defined by Wilson inclusion matrices $W_{t,n,k}$, proving exact values for all $t\le3$ at large $n$ and establishing tight bounds via design-theoretic and ILP methods. It recasts the problem in terms of binary $t$-designs and leverages reduced incidence matrices, derivative designs, and an ILP framework to bound the minimal block counts in $t$-designs, with detailed results for $t=0,1,2,3$ and both parity classes of $k$. The paper also connects these combinatorial structures to LDPC and locally recoverable codes, and it demonstrates the construction of quasi-cyclic LDPC codes from Wilson matrices with competitive decoding performance. Computational techniques (including MAGMA-based ILP searches) are used to sharpen known bounds and produce concrete reduced-incidence realizations, highlighting a productive interplay between combinatorial design theory and coding theory.

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.

Binary codes from subset inclusion matrices

TL;DR

This work determines the minimum distance for binary codes defined by Wilson inclusion matrices , proving exact values for all at large and establishing tight bounds via design-theoretic and ILP methods. It recasts the problem in terms of binary -designs and leverages reduced incidence matrices, derivative designs, and an ILP framework to bound the minimal block counts in -designs, with detailed results for and both parity classes of . The paper also connects these combinatorial structures to LDPC and locally recoverable codes, and it demonstrates the construction of quasi-cyclic LDPC codes from Wilson matrices with competitive decoding performance. Computational techniques (including MAGMA-based ILP searches) are used to sharpen known bounds and produce concrete reduced-incidence realizations, highlighting a productive interplay between combinatorial design theory and coding theory.

Abstract

In this paper, we study the minimum distances of binary linear codes with parity check matrices formed from subset inclusion matrices , representing -element subsets versus -element subsets of an -element set. We provide both lower and upper bounds on the minimum distances of these codes and determine the exact values for any and sufficiently large . Our study combines design and integer linear programming techniques. The codes we consider are connected to locally recoverable codes, LDPC codes and combinatorial designs.
Paper Structure (14 sections, 7 theorems, 46 equations, 2 figures, 1 table)

This paper contains 14 sections, 7 theorems, 46 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Linear codes
  4. Designs
  5. Constructions of binary designs
  6. Lower bounds on the number of blocks in binary designs
  7. Minimum number of blocks in a binary $2$-design
  8. Minimum number of blocks in a $3$-$(n,k)$-design for odd $k$
  9. Minimum number of blocks in a $3$-$(n,k)$-design for even $k$
  10. The properties of reduced incidence matrices
  11. Integer linear programming for $3$-$(n,k)$-designs
  12. ILP-based search of reduced incidence matrices of $3$-$(n,k)$-designs for even $k$
  13. Quasicyclic LDPC codes from Wilson inclusion matrices
  14. MAGMA program for finding binary $3$-$(n,k)$-designs with small number of blocks

Key Result

Theorem 1

WZ, Pot The minimum distance $d_{k-1,n,k}$ of the code $C_{k-1,n,k}$ is $k+1$.

Figures (2)

  • Figure 1: Decoding performance of QC LDPC codes from Wilson matrices vs MakKay type 1A code under Multi GBDF decoder with 30 iterations max in binary symmetric channel.
  • Figure 2: Decoding performance of QC LDPC codes from Wilson matrices vs MakKay type 1A code under layered min sum algorithm with 30 iterations max in AWGN channel.

Theorems & Definitions (9)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • Corollary 1
  • Theorem 4
  • Theorem 5
  • Theorem 6