On the Pseudocodeword Redundancy of Binary Linear Codes
The AWGNC, BSC, and max-fractional pseudocode-word redundancies of a binary linear code are defined to be the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudo-weight is equal to the minimum Hamming distance of the code. It is shown that most codes do not have a finite pseudocode-word redundancy. Also, upper bounds on the pseudocode-word redundancy for some families of codes, including codes based on designs, are provided. The pseudocode-word redundancies for all codes of small length (at most 9) are computed. Furthermore, comprehensive results are provided on the cases of cyclic codes of length at most 250 for which the eigenvalue bound of Vontobel and Koetter is sharp.