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

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