Abstract-The role of pseudocodewords in causing noncodeword outputs in linear programming (LP) decoding, graph cover decoding, and iterative message-passing decoding is investigated. The three main types of pseudocodewords in the literature -linear programming pseudocodewords, graph cover pseudocodewords, and computation tree pseudocodewords -are reviewed and connections between them are explored. Some discrepancies in the literature on minimal and irreducible pseudocodewords are highlighted and clarified, and a value for the minimal degree cover necessary to realize an LP pseudocodeword is found. Additionally, some conditions for the existence of connected realizations of graph cover pseudocodewords are given. This allows for further analysis of when graph cover pseudocodewords induce computation tree pseudocodewords. Finally, an example is offered that shows that existing theories on the distinction between graph cover pseudocodewords and computation tree pseudocodewords are incomplete.

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Abstract-Simulations have shown that the outputs of minsum (MS) decoding generally behave in one of two ways: the output either eventually stabilizes at a codeword or eventually cycles through a finite set of vectors that may include both codewords and non-codewords. This inconsistency in MS across iterations has significantly contributed to the difficulty in studying the performance of this decoder. To overcome this problem, a new decoder, average min-sum (AMS), is proposed; this decoder outputs the average of the min-sum output vectors over a finite set of iterations. Simulations comparing MS, AMS, linear programming (LP) decoding, and maximum likelihood (ML) decoding are presented, illustrating the relative performances of each of these decoders. In general, MS and AMS have comparable word error rates, and in the simulation most resembling codes of practical interest AMS is shown to have significantly lower bit error rate, demonstrating the potential benefits of this decoder in its own right. Additionally, the performance of MS and AMS relative to ML and LP decoding is consistent across simulations, indicating that AMS is a valid and potentially important tool for better analyzing MS performance and its relationship to other decoders. Finally, AMS pseudocodewords are introduced and analyzed and their relationship to graph cover and LP pseudocodewords is explored, with particular focus on the AMS pseudocodewords of regular LDPC codes and cycle codes.

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