This paper investigates both the algebraic, and combinatorial properties of these constructions of graph codes and digraph codes and shows that they possess the same desirable attributes as Cayley codes, but with added freedom.
Abstract
In 2012, Kaufman and Lubotzky constructed the first family of symmetric LDPC good codes. Their construction used Cayley codes, as originally defined by Kaufman and Wigderson (2016). In this paper we present two generalisations to the Cayley code construction, which we call graph codes and digraph codes. We investigate both the algebraic, and combinatorial properties of these constructions and show that they possess the same desirable attributes as Cayley codes, but with added freedom. We analyse the relationship between the expansion properties of the ingredient (di)graphs and the parameters of the constructed codes; our analysis offers an improvement to the results of Kaufman and Lubotzky. As an application, we construct an infinite family of good digraph codes, and we propose a series of open problems.
In this paper, we present a method for constructing linear codes derived from book graphs. Applying such a method, we construct binary codes that have the greatest possible minimum distance for their length and dimension by means of the $2$-sum of two equal book graphs. We then utilize matroid and graph theory together...
A. S. Barbosa, Franklin de L. Marquezino, G. L. La Guardia· 0 citations
This paper focuses on the construction and analysis of codes over the Frobenius ring T. By exploiting the idempotent structure of the ring, codes over T are decomposed into direct sums of component codes over a related local Frobenius non-chain ring. A gray map is constructed and shown to be bijective and distance-pres...
Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge...
Sara Asensio, Giulia Gaggero, Naveena Ragunathan et al.· 0 citations
Linear complementary dual (LCD) codes form an important class of linear codes with applications in cryptography, classical error correction, and quantum coding theory. In this paper, we study the neighbor relation on LCD codes over finite fields and the graph induced by this relation, where two codes are adjacent whene...
Javier de la Cruz, A. Horlemann, Marc Newman et al.· 0 citations
We give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of Kalachev and Panteleev (FOCS, 2025), does not rely on explicit constructions of asymptotically optimal locally testable codes ($c^3$-LTCs). For every fixed number of component codes and dimensions whose rates are b...
Eshan Chattopadhyay, Noam Ringach, Nicholas Spooner· 0 citations
We introduce a new family of recursively constructed sparse matrices, termed Fractus matrices, and investigate their use in constructing low-density parity-check (LDPC) codes. Generated through a self-similar recursive process, these matrices yield regular sparse parity-check matrices while preserving key structural pr...
Jesús Carrillo-Pacheco· 0 citations
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