Pseudocodewords, and in particular minimal pseudocodewords, play an important role in understanding the performance of linear programming (LP) decoding. In this paper, we investigate minimal pseudocodewords of binary Hamming codes described by full-rank parity-check matrices. We first provide some general results on minimal pseudocodewords with support size 3 of a binary parity-check matrix. We also prove a lower bound on the minimum binary symmetric channel (BSC) pseudoweight of a binary parity-check matrix. Then we prove that a full-rank parity-check matrix of a binary Hamming code has minimal pseudocodewords of certain types whose support sizes are larger than 3. Interestingly enough, the BSC pseudoweight of all these minimal pseudocodewords is 2. Using this fact as well as the above-mentioned lower bound, we further prove that a full-rank parity-check matrix of a binary Hamming code has minimum BSC pseudoweight 2. Moreover, the additive white Gaussian noise channel (AWGNC) pseudoweight of all these minimal pseudocodewords is 3. Based on numerical observations, we conjecture that a full-rank parity-check matrix of a binary Hamming code has minimum AWGNC pseudoweight 3. Finally, we provide more properties of a subset of minimal pseudocodewords of a full-rank parity-check matrix of a binary Hamming code.

The spark is an important property that provides recovery guarantees of a measurement matrix in compressed sensing. In this letter, we focus on a specific class of measurement matrices, i.e., binary low-density parity-check (LDPC) matrices from complete protographs. An upper bound on the spark of a binary LDPC measurement matrix from complete protograph is provided. In addition, the spark of array-based LDPC measurement matrices with certain column weights is analyzed.

Absorbing sets, a class of combinatorial structures of the Tanner graph representation of a low-density parity-check (LDPC) code, are known to influence the performance of the code under message passing iterative decoding. In this paper, we study the smallest absorbing sets of LDPC codes constructed from projective planes and Euclidean planes. The lower bounds on the parameters of smallest absorbing sets given by Dolecek are proven to be tight for these two families of LDPC codes. We also analyze the combinatorial properties of the smallest absorbing sets and give conditions necessary and sufficient for a set of bit nodes in the Tanner graph to be a smallest absorbing set. For LDPC codes from projective planes, we further give a condition necessary and sufficient for a smallest absorbing set to be a fully absorbing set. In addition, we show that these smallest absorbing sets are asymptotically not stable, which may explain to some extent the good performance as well as the low error floor expectation of these two families of LDPC codes.