Combinatorial Analysis of Dyadic and Quasi-Dyadic Codes

TL;DR

This paper introduces an algebraic framework for constructing and analyzing dyadic and quasi-dyadic quantum LDPC codes, focusing on cycle enumeration, girth maximization, and error-floor mitigation.

Contribution

It develops methods to control short cycles in dyadic LDPC codes, introduces dyadic-aware construction algorithms, and demonstrates decoding improvements through simulations.

Findings

Efficient enumeration of short cycles in Tanner graphs.

Dyadic-aware algorithms can maximize girth or minimize short cycle multiplicity.

Reducing short-cycle multiplicity improves decoding performance in simulations.

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.