Pseudocodewords of Parity-Check Codes

The success of modern algorithms for the decoding problem such as message-passing iterative decoding and linear programming decoding lies in their local nature. This feature allows the algorithms to be extremely fast and capable of correcting more errors than guaranteed by the classical minimum distance of the code. Nonetheless, the performance of these decoders depends crucially on the Tanner graph representation of the code. In order to understand this choice of representation, we need to analyze the pseudocodewords of the Tanner graph of a code. These pseudocodewords are outputs of local decoding algorithms which may not be legitimate codewords. In this dissertation, we introduce a lifted fundamental cone and show that there is a one-to-one correspondence between graph cover pseudocodewords of a binary code and integer points in the lifted fundamental cone. We use this fact to prove the rationality of the generating function of the pseudocodewords for a general binary parity-check code. Our approach also yields algorithms for producing this generating function and provides tools for studying the irreducible pseudocodewords. Understanding irreducible pseudocodewords is crucial to determining the best representation of a code. Moreover, combining these techniques with the recent characterization of fundamental cone over F_3, we can analyze ternary parity-check codes. Finally, we make progress in the study of more general nonbinary codes by determining constraints satisfied by all pseudocodewords of a code over F_p where p is prime

On Pseudocodewords and Improved Union Bound of Linear Programming Decoding of HDPC Codes

In this paper, we present an improved union bound on the Linear Programming (LP) decoding performance of the binary linear codes transmitted over an additive white Gaussian noise channels. The bounding technique is based on the second-order of Bonferroni-type inequality in probability theory, and it is minimized by Prim's minimum spanning tree algorithm. The bound calculation needs the fundamental cone generators of a given parity-check matrix rather than only their weight spectrum, but involves relatively low computational complexity. It is targeted to high-density parity-check codes, where the number of their generators is extremely large and these generators are spread densely in the Euclidean space. We explore the generator density and make a comparison between different parity-check matrix representations. That density effects on the improvement of the proposed bound over the conventional LP union bound. The paper also presents a complete pseudo-weight distribution of the fundamental cone generators for the BCH[31,21,5] code