Constructing linear codes from digraphs and groups

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.