Efficiently Decoding Reed-Muller Codes From Random Errors

Reed-Muller (RM) codes encode an m-variate polynomial of degree at most r by evaluating it on all points in {0,1} m . We denote this code by RM(r,m). The minimum distance of RM(r,m) is 2 m-r and so it cannot correct more than half that number of errors in the worst case. For random errors one may ho...

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Vydané v:IEEE transactions on information theory Ročník 63; číslo 4; s. 1954 - 1960
Hlavní autori: Saptharishi, Ramprasad, Shpilka, Amir, Volk, Ben Lee
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York IEEE 01.04.2017
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:
Algorithms
Capacity planning
Codes
Computer science
Decoding
decoding algorithms
Electronic mail
Entropy
Error correction
Linear codes
Linear equations
Polynomials
Random errors
Reed-Muller codes
Reed-Solomon codes
ISSN:0018-9448, 1557-9654
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Shrnutí:Reed-Muller (RM) codes encode an m-variate polynomial of degree at most r by evaluating it on all points in {0,1} m . We denote this code by RM(r,m). The minimum distance of RM(r,m) is 2 m-r and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this paper we give an efficient algorithm (in the block length n=2 m ) for decoding random errors in RM codes far beyond the minimum distance. Specifically, for low-rate codes (of degree r=o(√m)), we can correct a random set of (1/2-o(1))n errors with high probability. For high rate codes (of degree m-r for r=o(√m/log m)), we can correct roughly m r/2 errors. More generally, for any integer r, our algorithm can correct any error pattern in RM(m-(2r+2),m), for which the same erasure pattern can be corrected in RM(m-(r+1),m). The results above are obtained by applying recent results of Abbe, Shpilka, and Wigderson (STOC, 2015) and Kudekar et al. (STOC, 2016) regarding the ability of RM codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding RM codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the error-locating pair method of Pellikaan, Duursma, and Kötter that generalizes the Berlekamp-Welch algorithm for decoding Reed-Solomon codes.
Bibliografia:ObjectType-Article-1
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content type line 14
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2017.2671410