Combinatorial Analysis of Dyadic and Quasi-Dyadic Codes

2026-05-03

AI Summary

This work addresses the susceptibility of quantum low-density parity-check (QLDPC) codes to harmful structures—particularly short cycles in their Tanner graphs—that degrade iterative decoding performance. Leveraging the recursive block structure of binary dyadic and quasi-dyadic matrices, the authors develop an algebraic framework that characterizes and controls short cycles in lifted graphs through tailless, backtrackless closed walks in the base protograph. They propose a novel dyadic-aware PEG-like construction algorithm that employs forbidden shift sets to either maximize girth or minimize the multiplicity of shortest cycles. For the first time, they explicitly enumerate absorbing sets under dyadic layout constraints and devise a method to construct dyadic QLDPC codes satisfying the CSS commutativity conditions. Simulations demonstrate that significantly reducing the number of short cycles yields notable decoding gains, even when girth cannot be increased, thereby validating the efficacy of the proposed approach.

Abstract

Quantum low-density parity-check (QLDPC) codes offer a promising route to scalable fault-tolerant quantum computation, but their performance under iterative decoding is strongly influenced by short-cycle structure and other harmful subgraphs in the associated Tanner graphs. This paper develops an algebraic framework for constructing and analyzing (Q)LDPC codes from dyadic and quasi-dyadic matrices-translation-invariant $2^\ell \times 2^\ell$ binary matrices specified compactly by a signature row and forming a commutative ring with recursive block structure. Leveraging this structure, we relate cycles in lifted Tanner graphs to tailless backtrackless closed walks in the protograph and derive efficient, implementable methods to enumerate and control short cycles (notably $4$-, $6$-, and $8$-cycles). We introduce dyadic-aware PEG-style construction algorithms that use forbidden sets of shifts to maximize attainable girth when possible and otherwise minimize the multiplicity of the shortest cycles at the target girth. Motivated by error-floor phenomena, we further characterize and explicitly enumerate absorbing sets in key dyadic layout boundary cases, identifying configurations that induce abundant $(a,0)$-absorbing sets. Finally, we propose CSS-oriented dyadic constructions that satisfy commutation constraints by design and demonstrate via belief-propagation simulations that reducing short-cycle multiplicity can yield substantial decoding gains even when girth cannot be increased.

Problem

• QLDPC codes
• short cycles
• Tanner graphs
• absorbing sets
• iterative decoding

Innovation

• dyadic codes
• short-cycle enumeration
• absorbing sets
• QLDPC construction
• PEG algorithm