American Institute of Mathematical Sciences

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

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

José Gómez-Torrecillas 1, , F. J. Lobillo 1, and  Gabriel Navarro 2,

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.
Keywords: block error--correcting code, information--bit error rate (iBER), bit--error rate (BER), MDS code, false positive..
Mathematics Subject Classification: Primary: 94B70, 94B0.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

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