Estimating the fraction of erasure patterns correctable by linear codes
The conditional probability (fraction) of the successful decoding of erasure patterns of high (greater than the code distance) weights is investigated for linear codes with the partially known or unknown weight spectra of code words. The estimated conditional probabilities and the methods used to calculate them refer to arbitrary binary linear codes and binary Hamming, Panchenko, and Bose–Chaudhuri–Hocquenghem (BCH) codes, including their extended and shortened forms. Error detection probabilities are estimated under erasure-correction conditions. The product-code decoding algorithms involving the correction of high weight erasures by means of component Hamming, Panchenko, and BCH codes are proposed, and the upper estimate of decoding failure probability is presented.
Keywordslinear code erasure correction product code
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- 3.M. N. Nazarov and S. P. Mishin, “Almost optimum codes used to correct erasures,” in Vestn. Volgograd. Gos. Univ., Ser. 1: Mat., Fiz., No. 4, 59–69 (1999) [in Russian].Google Scholar
- 5.O. V. Popov, “On assessment of linear code ability to correct erasures and find errors in the presence of erasures,” Elektrosvyaz’, No. 10, 1 (1967) [in Russian].Google Scholar
- 6.O. V. Popov, On Correction of Erasures by Cyclic Codes. Digital Data Transmission over Channels with Memory (Nauka, Moscow, 1970), pp. 111–124 [in Russian].Google Scholar
- 9.A. A. Davydov, A. Yu. Drozhzhina-Labinskaya, and L. M. Tombak, “Supplementary correcting possibilities of BCH codes correcting double errors and finding triple errors,” in Cybernetics questions. Complex design of element and design base of the supercomputer, by Ed. V. A. Mel’nikov, and Yu. I. Mitropol’skii (VINITI, Moscow, 1988), pp. 86–112 [in Russian].Google Scholar
- 13.V. I. Panchenko, “On optimization of a linear code with distance 4,” in Proc. VIII All-Union Conf. on According to the Theory of Coding and Information Transfer, Kuibyshev, 1981, Part 2: The Theory of Coding (Akad. Nauk SSSR, Moscow, 1981), pp. 132–134 [in Russian].Google Scholar
- 15.K. M. Cheung, “The weight distribution and randomness of linear codes,” in TDA Progress Report 42-97, Jet Propulsion Lab., Pasadena, CA, California, USA, 1989 (Inst. of Tech., Pasadena, 1989), pp. 208–215. https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19890018521.pdf. Accessed 22.11.2016.Google Scholar
- 19.R. H. Morelos-Zaragoza, The Art of Error Correcting Coding (Wiley, Chichester, 2002; Tekhnosfera, Moscow, 2005).Google Scholar
- 21.Weight Distribution. http://www.ec.okayama-u.ac.jp//~infsys/kusaka/wd/index.html (Accessed 27.11.2016).Google Scholar