# American Institute of Mathematical Sciences

February  2016, 10(1): 11-28. doi: 10.3934/amc.2016.10.11

## An approach to the performance of SPC product codes on the erasure channel

 1 Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas (UNICAMP), R. Sérgio Buarque de Holanda, 651, Cidade Universitária, Campinas - SP, 13083-859, Brazil 2 Departament de Matemàtiques, Universitat d'Alacant, Ap. Correus 99, E-03080, Alacant, Spain

Received  November 2014 Revised  September 2015 Published  March 2016

Product codes can be used to correct errors or recover erasures. In this work we consider the simplest form of a product code, this is, the single parity check (SPC) product code. This code has a minimum distance of four and is thus guaranteed to recover all single, double, and triple erasure patterns. The code is actually capable of recovering a higher number of erasure patterns. We count the number of uncorrectable erasure patterns of size $n \times n$ with $t$ erasures, for $t=8$, $2n-3$, $2n-2$ and $2n-1$, using the relation between erasure patterns and bipartite graphs.
Citation: Sara D. Cardell, Joan-Josep Climent. An approach to the performance of SPC product codes on the erasure channel. Advances in Mathematics of Communications, 2016, 10 (1) : 11-28. doi: 10.3934/amc.2016.10.11
##### References:
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##### References:
 [1] M. Z. Abu-Sbeih, On the number of spanning trees of $K_n$ and $K_{m,n}$,, Discrete Math., 84 (1990), 205. doi: 10.1016/0012-365X(90)90377-T. [2] R. Amutha, K. Verraraghavan and S. K. Srivatsa, Recoverability study of SPC product codes under erasure decoding,, Inf. Sci., 173 (2005), 169. doi: 10.1016/j.ins.2004.07.011. [3] R. Diestel, Graph Theory,, Springer-Verlag, (2000). doi: 10.1007/b100033. [4] P. Elias, Coding for noisy channels,, IRE International Convention Record, (1955), 37. [5] P. Giblin, Graphs, Surfaces and Homology, 3rd edition,, Cambridge Univ. Press, (2010). doi: 10.1017/CBO9780511779534. [6] N. Hartsfield and J. S. Werth, Spanning trees of the complete bipartite graph,, in Topics in Combinatorics and Graph Theory (eds. R. Bodendieck and R. Henn), (1990), 339. [7] M. A. Kousa, A novel approach for evaluating the performance of SPC product codes under erasure decoding,, IEEE Trans. Commun., 50 (2002), 7. [8] M. A. Kousa and A. H. Mugaibel, Cell loss recovery using two-dimensional erasure correction for ATM networks,, in Proc. 7th Int. Conf. Telecommun. Syst., (1999), 85. [9] A. Muqaibel, Enhanced upper bound for erasure recovery in SPC product codes,, ETRI J., 31 (2009), 518. [10] D. M. Rankin and T. A. Gulliver, Single parity check product codes,, IEEE Trans. Commun., 49 (2001), 1354. [11] J. M. Simmons and R. G. Gallager, Design of error detection scheme for class C service in ATM,, IEEE/ACM Trans. Netw., 2 (1994), 80.
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