On Codes with Support-Constrained Parity Checks

TL;DR

This paper investigates the maximum achievable minimum distance of linear codes under support constraints on the parity-check matrix, providing theoretical bounds, constructions, and practical insights.

Contribution

It derives optimal distance bounds under support constraints, shows limitations of Reed--Solomon subcodes in the parity-check setting, and analyzes structured constraints for code design.

Findings

Optimal minimum distance bounds are derived for support-constrained codes.

Reed--Solomon subcodes cannot always achieve the optimal distance under support constraints.

Structured constraint families are analyzed to guide practical code construction.

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.