The Stopping Redundancy Hierarchy of Cyclic Codes

We extend the framework for studying the stopping redundancy of a linear block code by introducing and analyzing the stopping redundancy hierarchy. The stopping redundancy hierarchy of a code represents a measure of the trade-off between performance and complexity of iteratively decoding a code used over the binary erasure channel. It is defined as an ordered list of positive integers in which the ith entry, termed the i-th stopping redundancy, corresponds to the minimum number of rows in any parity-check matrix of the code that has stopping distance at least i. In particular, we derive lower and upper bounds for the i-th stopping redundancy of a code by using probabilistic methods and Bonferroni-type inequalities. Furthermore, we specialize the findings for cyclic codes, and show that parity-check matrices in cyclic form have some desirable redundancy properties. We also propose to investigate the influence of the generator codeword of the cyclic parity-check matrix on its stopping distance properties.