4 research outputs found
Efficiently decoding Reed-Muller codes from random errors
Reed-Muller codes encode an -variate polynomial of degree by
evaluating it on all points in . We denote this code by .
The minimal distance of is 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 work we give an efficient algorithm (in the block length ) for
decoding random errors in Reed-Muller codes far beyond the minimal distance.
Specifically, for low rate codes (of degree ) we can correct a
random set of errors with high probability. For high rate codes
(of degree for ), we can correct roughly
errors.
More generally, for any integer , our algorithm can correct any error
pattern in for which the same erasure pattern can be corrected
in . The results above are obtained by applying recent results
of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and
Kudekar et al. (2015) regarding the ability of Reed-Muller 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
Reed-Muller 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 famous Berlekamp-Welch algorithm for decoding
Reed-Solomon codes.Comment: 18 pages, 2 figure
On an Asymptotic Series of Ramanujan
An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is
concerned with the behavior of the expected value of for large
where is a Poisson random variable with mean and
is a function satisfying certain growth conditions. We generalize this by
studying the asymptotics of the expected value of when the
distribution of belongs to a suitable family indexed by a convolution
parameter. Examples include the problem of inverse moments for distribution
families such as the binomial or the negative binomial.Comment: To appear, Ramanujan