LDPC Codes Based on Latin Squares: Cycle Structure, Stopping, and Trapping Set Analysis

2007 IEEE Transactions on Communications  
It is well known that certain combinatorial structures in the Tanner graph of a low-density parity-check code exhibit a strong influence on its performance under iterative decoding. These structures include cycles, stopping/trapping sets and parameters such as the diameter of the code. In general, it is very hard to find a complete characterization of such configurations in an arbitrary code, and even harder to understand the intricate relationships that exist between these entities. It is
more » ... fore of interest to identify a simple setting in which all the described combinatorial structures can be enumerated and studied within a joint framework. One such setting is developed in this paper, for the purpose of analyzing the distribution of short cycles and the structure of stopping and trapping sets in Tanner graphs of LDPC codes based on idempotent and symmetric Latin squares. The parity-check matrices of LDPC codes based on Latin squares have a special form that allows for connecting combinatorial parameters of the codes with the number of certain sub-rectangles in the Latin squares. Sub-rectangles of interest can be easily identified, and in certain instances, completely enumerated. The presented study can be extended in several different directions, one of which is concerned with modifying the code design process in order to eliminate or reduce the number of configurations bearing a negative influence on the performance of the code. Another application of the results includes determining to which extent a configuration governs the behavior of the bit error rate (BER) curve in the waterfall and error-floor regions.
doi:10.1109/tcomm.2006.887791 fatcat:4c6fxug2fnbtnneq7mlilcmgyu