May  2015, 9(2): 149-168. doi: 10.3934/amc.2015.9.149

Information--bit error rate and false positives in an MDS code

1. 

Department of Algebra and CITIC-UGR, University of Granada, E18071 Granada, Spain, Spain

2. 

Department of Computer Sciences and AI, and CITIC-UGR, University of Granada, E18071 Granada, Spain

Received  July 2013 Published  May 2015

In this paper, a computation of the input-redundancy weight enumerator is presented. This is used to improve the theoretical approximation of the information--bit error rate, in terms of the channel bit--error rate, in a block transmission through a discrete memoryless channel. Since a bounded distance reproducing encoder is assumed, we introduce the here-called false positive, a decoding failure with no information-symbol error, and we estimate the probability that this event occurs. As a consequence, a new performance analysis of an MDS code is proposed.
Citation: José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Information--bit error rate and false positives in an MDS code. Advances in Mathematics of Communications, 2015, 9 (2) : 149-168. doi: 10.3934/amc.2015.9.149
References:
[1]

C. Desset, B. Macq and L. Vandendorpe, Computing the word-, symbol-, and bit-error rates for block error-correcting codes, IEEE Trans. Commun., 52 (2004), 910-921. doi: 10.1109/TCOMM.2004.829509.

[2]

R. Dodunekova and S. M. Dodunekov, Sufficient conditions for good and proper error-detecting codes, IEEE Trans. Inf. Theory, 43 (1997), 2023-2026. doi: 10.1109/18.641570.

[3]

R. Dodunekova and S. M. Dodunekov, The MMD codes are proper for error detection, IEEE Trans. Inf. Theory, 48 (2002), 3109-3111. doi: 10.1109/TIT.2002.805082.

[4]

R. Dodunekova, S. M. Dodunekov and E. Nikolova, A survey on proper codes, Discrete Appl. Math., 156 (2008), 1499-1509. doi: 10.1016/j.dam.2005.06.014.

[5]

M. El-Khamy, New Approaches to the Analysis and Design of Reed-Solomon Related Codes, Ph.D thesis, California Institute of Technology, 2007.

[6]

M. El-Khamy and R. J. McEliece, Bounds on the average binary minimum distance and the maximum likelihood performance of Reed Solomon codes, in 42nd Allerton Conf. Commun. Control Comput., 2004.

[7]

M. El-Khamy and R. J. McEliece, The partition weight enumerator of MDS codes and its applications, in Int. Symp. Inf. Theory, 2005, 926-930. doi: 10.1109/ISIT.2005.1523473.

[8]

A. Faldum, J. Lafuente, G. Ochoa and W. Willems, Error probabilities for bounded distance decoding, Des. Codes Crypt., 40 (2006), 237-252. doi: 10.1007/s10623-006-0010-x.

[9]

M. P. C. Fossorier, Critical point for maximum likelihood decoding of linear block codes, IEEE Commun. Letters, 9 (2005), 817-819. doi: 10.1109/LCOMM.2005.1506713.

[10]

J. Han, P. H. Siegel and P. Lee, On the probability of undetected error for overextended Reed-Solomon codes, IEEE Trans. Inf. Theory, 52 (2006), 3662-3669. doi: 10.1109/ITW.2006.1633800.

[11]

T. Kasami and S. Lin, On the probability of undetected error for the maximum distance separable codes, IEEE Trans. Commun., COM-32 (1984), 998-1006. doi: 10.1109/TCOM.1984.1096175.

[12]

J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell System Tech. J., 42 (1963), 79-94. doi: 10.1002/j.1538-7305.1963.tb04003.x.

[13]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North Holland Publishing Co., 1977.

[14]

J. Riordan, Combinatorial Identities, Robert E. Krieger Publishing Co., Huntington, New York, 1979.

[15]

S. Roman, Coding and Information Theory, Springer-Verlag, New York, 1992.

[16]

W. A. Stein et al., Sage Mathematics Software (Version 5.9), The Sage Development Team, 2012, available at http://www.sagemath.org

[17]

D. Torrieri, The information-bit error rate for block codes, IEEE Trans. Commun., COM-32 (1984), 474-476. doi: 10.1109/TCOM.1984.1096082.

[18]

D. Torrieri, Information-bit, information-symbol, and decoded-symbol error rates for linear block codes, IEEE Trans. Commun., 36 (1988), 613-617. doi: 10.1109/26.1477.

[19]

J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd edition, Cambridge University Press, 2001. doi: 10.1017/CBO9780511987045.

[20]

K.-P. Yar, D.-S. Yoo and W. Stark, Performance of RS coded $M$-ary modulation with and without symbol overlapping, IEEE Trans. Commun., 56 (2008), 445-453. doi: 10.1109/TCOMM.2008.050229.

show all references

References:
[1]

C. Desset, B. Macq and L. Vandendorpe, Computing the word-, symbol-, and bit-error rates for block error-correcting codes, IEEE Trans. Commun., 52 (2004), 910-921. doi: 10.1109/TCOMM.2004.829509.

[2]

R. Dodunekova and S. M. Dodunekov, Sufficient conditions for good and proper error-detecting codes, IEEE Trans. Inf. Theory, 43 (1997), 2023-2026. doi: 10.1109/18.641570.

[3]

R. Dodunekova and S. M. Dodunekov, The MMD codes are proper for error detection, IEEE Trans. Inf. Theory, 48 (2002), 3109-3111. doi: 10.1109/TIT.2002.805082.

[4]

R. Dodunekova, S. M. Dodunekov and E. Nikolova, A survey on proper codes, Discrete Appl. Math., 156 (2008), 1499-1509. doi: 10.1016/j.dam.2005.06.014.

[5]

M. El-Khamy, New Approaches to the Analysis and Design of Reed-Solomon Related Codes, Ph.D thesis, California Institute of Technology, 2007.

[6]

M. El-Khamy and R. J. McEliece, Bounds on the average binary minimum distance and the maximum likelihood performance of Reed Solomon codes, in 42nd Allerton Conf. Commun. Control Comput., 2004.

[7]

M. El-Khamy and R. J. McEliece, The partition weight enumerator of MDS codes and its applications, in Int. Symp. Inf. Theory, 2005, 926-930. doi: 10.1109/ISIT.2005.1523473.

[8]

A. Faldum, J. Lafuente, G. Ochoa and W. Willems, Error probabilities for bounded distance decoding, Des. Codes Crypt., 40 (2006), 237-252. doi: 10.1007/s10623-006-0010-x.

[9]

M. P. C. Fossorier, Critical point for maximum likelihood decoding of linear block codes, IEEE Commun. Letters, 9 (2005), 817-819. doi: 10.1109/LCOMM.2005.1506713.

[10]

J. Han, P. H. Siegel and P. Lee, On the probability of undetected error for overextended Reed-Solomon codes, IEEE Trans. Inf. Theory, 52 (2006), 3662-3669. doi: 10.1109/ITW.2006.1633800.

[11]

T. Kasami and S. Lin, On the probability of undetected error for the maximum distance separable codes, IEEE Trans. Commun., COM-32 (1984), 998-1006. doi: 10.1109/TCOM.1984.1096175.

[12]

J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell System Tech. J., 42 (1963), 79-94. doi: 10.1002/j.1538-7305.1963.tb04003.x.

[13]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North Holland Publishing Co., 1977.

[14]

J. Riordan, Combinatorial Identities, Robert E. Krieger Publishing Co., Huntington, New York, 1979.

[15]

S. Roman, Coding and Information Theory, Springer-Verlag, New York, 1992.

[16]

W. A. Stein et al., Sage Mathematics Software (Version 5.9), The Sage Development Team, 2012, available at http://www.sagemath.org

[17]

D. Torrieri, The information-bit error rate for block codes, IEEE Trans. Commun., COM-32 (1984), 474-476. doi: 10.1109/TCOM.1984.1096082.

[18]

D. Torrieri, Information-bit, information-symbol, and decoded-symbol error rates for linear block codes, IEEE Trans. Commun., 36 (1988), 613-617. doi: 10.1109/26.1477.

[19]

J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd edition, Cambridge University Press, 2001. doi: 10.1017/CBO9780511987045.

[20]

K.-P. Yar, D.-S. Yoo and W. Stark, Performance of RS coded $M$-ary modulation with and without symbol overlapping, IEEE Trans. Commun., 56 (2008), 445-453. doi: 10.1109/TCOMM.2008.050229.

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