On the relevance of graph covers and zeta functions for the analysis of SPA decoding of cycle codes

For an arbitrary binary cycle code, we show that sum-product algorithm (SPA) decoding after infinitely many iterations equals symbolwise graph-cover decoding. We do this by characterizing the Bethe free energy function of the underlying normal factor graph (NFG) and by stating a global convergence proof of the SPA. We also show that the set of log-likelihood ratio vectors for which the SPA converges to the all-zero codeword is given by the region of convergence of the edge zeta function associated with the underlying NFG. The results in this paper justify the use of graph-cover pseudo-codewords and edge zeta functions to characterize the behavior of SPA decoding of cycle codes. These results have also implications for the analysis of attenuated sum-product and max-product algorithm decoding of low-density parity-check (LDPC) codes beyond cycle codes.

[1]  A. Terras,et al.  Zeta functions of finite graphs and coverings, III , 1996 .

[2]  K. Hashimoto.  Zeta functions of finite graphs and representations of p-adic groups , 1989 .

[3]  P. Vontobel,et al.  Graph-Cover Decoding and Finite-Length Analysis of Message-Passing Iterative Decoding of LDPC Codes , 2005, ArXiv.

[4]  S. Louis Hakimi,et al.  Graph theoretic error-correcting codes , 1968, IEEE Trans. Inf. Theory.

[5]  X. Jin.  Factor graphs and the Sum-Product Algorithm , 2002 .

[6]  Hilbert J. Kappen,et al.  Sufficient Conditions for Convergence of the Sum–Product Algorithm , 2005, IEEE Transactions on Information Theory.

[7]  A. Terras,et al.  Zeta Functions of Finite Graphs and Coverings , 1996 .

[8]  Pascal O. Vontobel,et al.  Counting in Graph Covers: A Combinatorial Characterization of the Bethe Entropy Function , 2010, IEEE Transactions on Information Theory.

[9]  Kenji Fukumizu,et al.  Graph Zeta Function in the Bethe Free Energy and Loopy Belief Propagation , 2009, NIPS.

[10]  Pascal O. Vontobel,et al.  The Bethe permanent of a non-negative matrix , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[11]  Rüdiger L. Urbanke,et al.  Modern Coding Theory , 2008 .

[12]  P. Vontobel,et al.  Graph-covers and iterative decoding of nite length codes , 2003 .

[13]  P. O. Vontobel,et al.  The Bethe Permanent of a Nonnegative Matrix , 2011, IEEE Transactions on Information Theory.

[14]  渡辺 有祐,et al.  Discrete geometric analysis of message passing algorithm on graphs , 2010 .

[15]  Guy Even,et al.  On Decoding Irregular Tanner Codes With Local-Optimality Guarantees , 2011, IEEE Transactions on Information Theory.

[16]  H.-A. Loeliger,et al.  An introduction to factor graphs , 2004, IEEE Signal Process. Mag..

[17]  Niclas Wiberg,et al.  Codes and Decoding on General Graphs , 1996 .

[18]  L. Litwin,et al.  Error control coding , 2001 .

[19]  P. Vontobel,et al.  Characterizations of pseudo-codewords of (low-density) parity-check codes , 2007 .

[20]  G. Forney,et al.  Codes on graphs: normal realizations , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[21]  Pascal O. Vontobel,et al.  Connecting the Bethe entropy and the edge zeta function of a cycle code , 2010, 2010 IEEE International Symposium on Information Theory.

[22]  Nathan Axvig,et al.  Graphical Characterizations of Linear Programming Pseudocodewords for Cycle Codes , 2013, IEEE Transactions on Information Theory.

[23]  Henry D. Pfister,et al.  Convergence of Weighted Min-Sum Decoding Via Dynamic Programming on Trees , 2011, IEEE Transactions on Information Theory.