Abstract
In the literature of Error Correcting Codes (ECC) there are many probabilistic characterizations of different events that happen in the decoding process. Historically, the most considered parameters in the performance of a linear code are the Probability of Undetected Error and the probability of incorrect decoding, also known as Probability of Miscorrected Error. While there is agreement about the first, starting from the Seventies, basically four definitions of the Probability of Miscorrected Error are present in literature; aim of this work is to show that they are equivalent and, although different in the mathematical formulation, they yield exactly the same result. The gap of this missing proof is herein fulfilled and two examples with interesting properties are given.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Biasi, N., Setti, F., Del Bue, A., Tavernini, M., Lunardelli, M., Fornaser, A., Da Lio, M., De Cecco, M.: Garment-based motion capture (gamocap): high-density capture of human shape in motion. Mach. Vis. Appl. 26(7–8), 955–973 (2015)
Blahut, R.E.: Theory and Practice of Error Control Codes. Addison-Wesley Pub. Co., Massachusetts (1983)
Caruso, F., Orsini, E., Sala, M., Tinnirello, C.: On the shape of the general error locator polynomial for cyclic codes. IEEE Trans. Inf. Theor. 63(6), 3641–3657 (2017)
Faldum, A., Lafuente, J., Ochoa, G., Willems, W.: Error probabilities for bounded distance decoding. Des. Codes Crypt. 40(2), 237–252 (2006)
Huntoon, Z., Michelson, A.: On the computation of the probability of post-decoding error events for block codes (corresp.). IEEE Trans. Inf. Theor. 23(3), 399–403 (1977)
Kim, M.G., Lee, J.H.: Decoder error probability of binary linear block codes and its application to binary primitive bch codes. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E79–A(4), 592–599 (1996)
McWilliams, F.J., Sloane, N.J.A.: The Theory of Error-correcting Codes. North-Holland, Amsterdam (1977)
Moon, T.K.: Error Correction Coding: Mathematical Methods and Algorithms. Wiley-Interscience, Hoboken (2005)
Piva, M., Sala, M.: A new bound for cyclic codes beating the roos bound. In: Muntean, T., Poulakis, D., Rolland, R. (eds.) CAI 2013. LNCS, vol. 8080, pp. 101–112. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40663-8_11
Roman, S.: Introduction to Coding and Information Theory. Undergraduate Texts in Mathematics. Springer, New York (1997)
Torrieri, D.J.: Principles of Spread-Spectrum Communication Systems. Springer, Cham (2005)
Vanstone, S.A., van Oorschot, P.C.: An Introduction to Error Correcting Codes with Applications. Kluwer International Series in Engineering and Computer Science: Communications and Information Theory. Springer, New York (1989)
Wicker, S.B.: Reed-solomon error control coding for rayleigh fading channels with feedback. IEEE Trans. Veh. Technol. 41(2), 124–133 (1992)
Wicker, S.B.: Error Control Systems for Digital Communication and Storage. Prentice Hall, New Jersey (1995)
Xia, S.-T., Fu, F.-W.: Undetected error probability of q-ary constant weight codes. Des. Codes Crypt. 48(2), 125–140 (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Frego, M. (2017). On the Probability of Incorrect Decoding for Linear Codes. In: O'Neill, M. (eds) Cryptography and Coding. IMACC 2017. Lecture Notes in Computer Science(), vol 10655. Springer, Cham. https://doi.org/10.1007/978-3-319-71045-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-71045-7_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-71044-0
Online ISBN: 978-3-319-71045-7
eBook Packages: Computer ScienceComputer Science (R0)