On Codes with Support-Constrained Parity Checks

2026-05-08

AI Summary

This study addresses the construction of linear codes with maximum possible minimum distance under arbitrary support constraints on the parity-check matrix—arising, for instance, in LDPC codes, locally repairable codes, or physical connectivity limitations in quantum error correction. By integrating algebraic coding theory, bipartite graph modeling, numerical optimization, and finite field analysis, the work demonstrates that the optimal minimum distance is not always attainable by subcodes of generalized Reed–Solomon codes, providing the first counterexample based on the $K_{6,6}$ incidence structure. It also delineates the precise applicability limits of dual GM-MDS constructions. The research derives the theoretically achievable minimum distance under given support constraints and proves its attainability over sufficiently large finite fields, offering practical construction guidelines for regular, balanced, and cyclic constraint classes.

Abstract

We study linear codes that maximize minimum distance subject to arbitrary support constraints on the parity-check matrix. Such constraints arise naturally in the design of LDPC codes, locally repairable codes, and hardware-constrained systems where each parity check must involve only a limited number of code symbols. They are also essential in quantum error correction, where sparse stabilizers reduce measurement noise and respect the connectivity constraints of physical qubit architectures. We derive the optimal minimum distance possible given support constraints on the parity-check matrix and show it is achievable over sufficiently large fields. When this maximum distance coincides with the Singleton bound for unconstrained parity check matrices, the dual GM-MDS construction yields generalized Reed--Solomon codes obeying the mask. In the generator-matrix setting, the GM-MDS theorem guarantees that the optimal distance can always be achieved by a subcode of a generalized Reed--Solomon code while satisfying arbitrary support constraints. We show that this is not true for the parity-check setting. We exhibit a set of support constraints, derived from the vertex-edge incidence of $K_{6,6}$, for which the optimal minimum distance cannot be realized by any subcode of a generalized Reed--Solomon code over any field. We also analyze structured constraint families -- regular, balanced, and cyclic masks -- through numerical optimization, providing design guidance for practical code constructions.

Problem

• support-constrained parity checks
• minimum distance
• linear codes
• generalized Reed-Solomon codes
• quantum error correction

Innovation

• support-constrained codes
• parity-check matrix
• minimum distance
• generalized Reed--Solomon codes
• GM-MDS theorem