Pseudocodewords, Expander Graphs, and the Algebraic Construction of Low-density Parity-check Codes

In 1948, Claude Shannon provided a sound mathematical foundation for communication, and proved the existence of good codes. Low-density parity-check (LDPC) codes are a family of error-correcting codes that achieve reliable performance (i.e., arbitrarily low probability of decoding error) near the Shannon-limit when decoded iteratively via very efficient graph-based message-passing decoders, the complexity of which grows only linearly in the block-length. Despite their tremendous success, LDPC codes lack a strong theoretical foundation and few explicit constructions of them are known that outperform their random counterparts. The first part of this thesis investigates how the structure of a graph representing an LDPC code affects the decoding performance. In particular, lower bounds on the minimum pseudocodeword weight, a key parameter in predicting iterative decoding performance, are obtained which link the performance of the code to the structural properties of its graph. One bound is a tree bound and is extended to generalized and p-ary LDPC codes. To this end, a suitable definition of pseudocodeword weight is derived for the p-ary symmetric channel. Moreover, the relationship between pseudocodeword weight distributions and different graph representations is examined. Bounds on the minimum pseudocodeword weight are also presented for several classes of expander codes based on the eigenvalues of the adjacency matrix of the Tanner graph. The second part of this thesis constructs LDPC codes having desirable properties. The first construction is combinatorial and uses permutations and mutually orthogonal Latin squares. One type results in the well-known projective-geometrybased LDPC codes. Another type yields a new family of codes having comparable parameters and pseudocodeword weights suitable for iterative decoding. The resulting performance of these codes is superior in comparison with that of comparable random LDPC codes, a feature attributed to the lack of low-weight bad pseudocodewords. For the second construction, unbalanced expander graphs are obtained by modifying the original zig-zag graph product to take bi-regular expander graphs as components, with suitable constraints on the degrees. Graphs resulting from the zig-zag and replacement product graphs are then used in the design of generalized LDPC codes having compact representation and pseudocodeword weights lower-bounded by the eigenvalue bounds.