TY - JOUR
T1 - LDPC codes based on latin squares
T2 - Cycle structure, stopping set, and trapping set analysis
AU - Laendner, Stefan
AU - Milenkovic, Olgica
N1 - Funding Information:
Paper approved by A. H. Banihashemi, the Editor for Coding and Communication Theory of the IEEE Communications Society. Manuscript received Spetember 21, 2004; revised February 3, 2006; April 28, 2006; May 1, 2006; and June 26, 2006. This work was supported in part by the National Science Foundation under Grant CCF-0514921. The work of S. Laendner was supported by a fellowship awarded by the Institute for Information Transmission (LIT) at the University of Erlangen-Nuremberg, Germany. This paper was presented in part at the IEEE International Conference on Communications, Paris, France, 2004.
PY - 2007/2
Y1 - 2007/2
N2 - It is well known that certain combinatorial structures in the Tanner graph of a low-density parity-check (LDPC) 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, therefore, 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 subrectangles in the Latin squares. Subrectangles of interest can be easily identified, and in certain instances, completely enumerated. This 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 curve in the waterfall and error-floor regions.
AB - It is well known that certain combinatorial structures in the Tanner graph of a low-density parity-check (LDPC) 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, therefore, 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 subrectangles in the Latin squares. Subrectangles of interest can be easily identified, and in certain instances, completely enumerated. This 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 curve in the waterfall and error-floor regions.
KW - Cayley Latin squares
KW - Design theory
KW - Low-density parity-check (LDPC) codes
KW - Stopping sets
KW - Trapping sets
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U2 - 10.1109/TCOMM.2006.888633
DO - 10.1109/TCOMM.2006.888633
M3 - Article
AN - SCOPUS:33947584908
SN - 0090-6778
VL - 55
SP - 303
EP - 312
JO - IEEE Transactions on Communications
JF - IEEE Transactions on Communications
IS - 2
ER -