| In order to communicate successfully over a noisy channel, a method for detecting and correcting transmission errors is needed. Coding theory is the study of efficient ways of encoding information so that error-detection and correction may be achieved. Low-density parity check codes possess excellent error-correcting capabilities while still operating quickly and efficiently.; Most constructions of LDPC codes have been through random search. Algebraic constructions allow us to analyze the codes in more detail. Using Ramanujan graphs and introducing the notion of a graph splitting, we describe a new construction to obtain families of LDPC codes. With the tools of Schubert calculus and Grassmannians we construct a new family of so-called finite geometry LDPC codes. In each case, we also describe the Tanner graph of the code.; Two important properties of the Tanner graph of an LDPC code are girth and expansion factor. For the graph theory constructions we analyze the Tanner graphs, computing girth and expansion factor, and compute the parameters of the codes. For the finite geometry construction, we compute the girth, and give a lower bound on the rate of the codes. |
