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We introduce new examples of Low Density Parity Check codes connected with the new families of regular graphs of bounded degree and increasing girth. Some new codes have an evident advantage in comparison with the D(n,q) based codes [9]. The new graphs are not edge transitive. So, they are not isomorphic to the Cayley graphs or those from the D(n,q) family, [14]. We use computer simulation to investigate spectral properties of graphs used for the construction of new codes. The experiment demonstrates existence of large spectral gaps in the case of each graph. We conjecture the existence of infinite families of Ramanujan graphs and expanders of bounded degree,existence of strongly Ramanujan graphs of unbounded degree. The lists of eigenvalues can be used for various practical applications of expanding graphs (Coding Theory, Networking, Image Processing). We show that new graphs can be used as a source of lists of cospectral pairs of graphs of bounded or unbounded degree.

In this paper we investigate correcting properties of LDPC codes obtained from families of algebraic graphs. The graphs considered in this article come from the infinite incidence structure. We describe how to construct these codes, choose the parameters and present several simulations, done by using the MAP decoder. We describe how error correcting properties are dependent on the graph structure. We compare our results with the currently used codes, obtained by Guinand and Lodge [1] from the family of graphs D(k; q), which were constructed by Ustimenko and Lazebnik [2].

In this paper we describe how to use special induced subgraphs of generalized m-gons to obtain the LDPC error correcting codes. We compare the properties of codes related to the affine parts of q-regular generalised 6-gons with the properties of known LDPC codes corresponding to the graphs D(5, q).

Expander graphs are highly connected sparse finite graphs. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. For practical applications it is very important to construct expander and Ramanujan graphs with given regularity and order. In general, constructions of the best expander graphs with a given regularity and order is no easy task. In this paper we present algorithms for generation of Ramanujan graphs and other expanders. We describe properties of obtained graphs in comparison to previously known results. We present a method to obtain a new examples of irregular LDPC codes based on described graphs and we briefly describe properties of this codes.