Heuristic Search for Minimum-Distance Upper-Bound Witnesses in Quantum APM-LDPC Codes

TL;DR

This paper develops a unified framework for constructing and verifying upper bounds on the minimum distance of quantum LDPC codes, improving bounds for specific code families using heuristic search and kernel tests.

Contribution

It introduces a novel, unified approach for certifying upper bounds on quantum LDPC code distances through candidate generation and kernel exclusion, with exact certification under certain conditions.

Findings

Sharpened upper bounds on quantum LDPC code distances.

Concrete certified bounds across various parameters.

Framework applicable to multiple code constructions.

Abstract

This paper investigates certified upper bounds on the minimum distance of an explicit family of Calderbank-Shor-Steane quantum LDPC codes constructed from affine permutation matrices. All codes considered here have active Tanner graphs of girth eight. Rather than attempting to prove a general lower bound for the full code distance, we focus on constructing low-weight non-stabilizer logical representatives, which yield valid upper bounds once they are verified to lie in the opposite parity-check kernel and outside the stabilizer row space. We develop a unified framework for such witnesses arising from latent row relations, restricted-lift subspaces including block-compressed, selected-fiber, and CRT-stripe constructions, cycle- 8 elementary trapping-set structures, and decoder-failure residuals. In every case, search is used only to generate candidates; the reported bounds begin only after explicit kernel and row-space exclusion tests have been passed. For the latent part, we also identify a block-compression criterion under which the certification becomes exact. Applying these methods to representative APM-LDPC codes sharpens previously reported upper bounds and provides concrete certified values across the explored parameter range.