arXiv subjects

Anthony Gómez-Fonseca

Publications and source records attributed to Anthony Gómez-Fonseca.

5 recordsLinked to original sources

arXiv2026

Low-Density Parity-Check Codes of High Girth from Permutation Polynomials

Quasi-cyclic low-density parity-check (QC-LDPC) codes are widely used in modern communication standards due to their simple structure and efficient encoder and decoder implementations. However, the commutativity of the circulant permutation matrices used in their construction significantly limits their girth and minimum distance, affecting iterative decoding performance. In this paper, we investigate protograph-based codes constructed using permutation polynomials of higher degree, focusing on a closed family of non-commuting binomial permutation polynomials that enables these constructions to exceed both of these girth and minimum distance upper bounds while maintaining a simple representation. Simulation results demonstrate improved decoding performance compared to constructions based on affine permutation polynomials, which include QC-LDPC codes.

cs.IT↗

arXiv2026

Fault-Tolerant QLDPC Syndrome Measurement via LDGM Encoding

We propose the use of certain low-density generator-matrix (LDGM) codes as syndrome measurement (SM) codes for quantum low-density parity check (QLDPC) codes. We use an efficient progressive-edge-growth-like algorithm to create LDGM SM codes with column and row weights that result in measured stabilizers that have constant weight, thus preserving the desirable properties of the underlying QLDPC code. This process allows for control over stabilizer weights and SM code distance, resulting in significantly better performance than repeated syndrome extraction and allowing for both higher distances and fewer syndrome measurements. We implement these SM codes on a distance-5 rotated surface code, and show that this procedure results in a lower probability of logical error. As syndrome measurements performed are a reasonable metric for the time a circuit takes to implement, we conclude that these LDGM codes allow for improved implementation of QLDPC codes without sacrificing the low weights of the syndrome measurements performed.

quant-ph↗

arXiv2026

Combinatorial Analysis of Dyadic and Quasi-Dyadic Codes

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.

cs.IT↗

arXiv2025

Generalized Quasi-Cyclic LDPC Codes: Design and Efficient Encoding

Generalized low-density parity-check (GLDPC) codes, where single parity-check constraints on the code bits are replaced with generalized constraints (an arbitrary linear code), are a promising class of codes for low-latency communication. The block error rate performance of the GLDPC codes, combined with a complementary outer code, has been shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G ultra-reliable low-latency communication (URLLC) regime. A major drawback of these codes is that it is not known how to construct appropriate polynomial matrices to encode them efficiently. In this paper, we analyze practical constructions of quasi-cyclic GLDPC (QC-GLDPC) codes and show how to construct polynomial generator matrices in various forms using minors of the polynomial matrix. The approach can be applied to fully generalized matrices or partially generalized (with mixed constraint node types) to find better performance/rate trade-offs. The resulting encoding matrices are presented in useful forms that facilitate efficient implementation. The rich substructure displayed also provides us with new methods of determining low weight codewords, providing lower and upper bounds on the minimum distance and often giving those of weight equal to the minimum distance. Based on the minors of the polynomial parity-check matrix, we also give a formula for the rank of any parity-check matrix representing a QC-LDPC or QC-GLDPC code, and hence, the dimension of the code. Finally, we show that by applying double graph-liftings, the code parameters can be improved without affecting the ability to obtain a polynomial generator matrix.

cs.IT↗

arXiv2021

Necessary and Sufficient Girth Conditions for LDPC Tanner Graphs with Denser Protographs

This paper gives necessary and sufficient conditions for the Tanner graph of a quasi-cyclic (QC) low-density parity-check (LDPC) code based on the all-one protograph to have girth 6, 8, 10, and 12, respectively, in the case of parity-check matrices with column weight 4. These results are a natural extension of the girth results of the already-studied cases of column weight 2 and 3, and it is based on the connection between the girth of a Tanner graph given by a parity-check matrix and the properties of powers of the product between the matrix and its transpose. The girth conditions can be easily incorporated into fast algorithms that construct codes of desired girth between 6 and 12; our own algorithms are presented for each girth, together with constructions obtained from them and corresponding computer simulations. More importantly, this paper emphasizes how the girth conditions of the Tanner graph corresponding to a parity-check matrix composed of circulants relate to the matrix obtained by adding (over the integers) the circulant columns of the parity-check matrix. In particular, we show that imposing girth conditions on a parity-check matrix is equivalent to imposing conditions on a square circulant submatrix of size 4 obtained from it.

cs.IT↗