Analysis and Enumeration of Absorbing Sets for Non-Binary Graph-Based Codes

In this work, we first provide the definition of absorbing sets for linear channel codes over non-binary alphabets. In a graphical representation of a non-binary channel code, an absorbing set can be described by a collection of topological and edge labeling conditions. In the non-binary case, the equations relating neighboring variable and check nodes are over a non-binary field, and the edge weights are given by the non-zero elements of that non-binary field. As a consequence, it becomes more difficult for a given structure to satisfy the absorbing set constraints compared to the binary case. This observation in part explains the superior performance of non-binary codes over their binary counterparts. We show that the conditions in the non-binary absorbing set definition can be simplified in the case of non-binary elementary absorbing sets. Based on these simplified conditions, we provide design guidelines for finite-length non-binary codes free of small non-binary elementary absorbing sets. These guidelines demonstrate that even under the preserved topology, the performance of a non-binary graph-based code in the error floor region can be substantially improved by manipulating edge weights so as to avoid small absorbing sets. Our various simulation results suggest that the proposed non-binary absorbing set definition is useful for a range of code constructions and decoders. Finally, by using both insights from graph theory and combinatorial techniques, we establish the asymptotic distribution of non-binary elementary absorbing sets for regular code ensembles.