21 research outputs found
List decoding Reed-Muller codes over small fields
The list decoding problem for a code asks for the maximal radius up to which
any ball of that radius contains only a constant number of codewords. The list
decoding radius is not well understood even for well studied codes, like
Reed-Solomon or Reed-Muller codes.
Fix a finite field . The Reed-Muller code
is defined by -variate degree-
polynomials over . In this work, we study the list decoding radius
of Reed-Muller codes over a constant prime field ,
constant degree and large . We show that the list decoding radius is
equal to the minimal distance of the code.
That is, if we denote by the normalized minimal distance of
, then the number of codewords in any ball of
radius is bounded by independent
of . This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008],
who among other results proved it in the special case of
; and extends the work of Gopalan [FOCS 2010] who
proved the conjecture in the case of .
We also analyse the number of codewords in balls of radius exceeding the
minimal distance of the code. For , we show that the number of
codewords of in a ball of radius is bounded by , where
is independent of . The dependence on is tight.
This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who
proved similar bounds over .
The proof relies on several new ingredients: an extension of the
Frieze-Kannan weak regularity to general function spaces, higher-order Fourier
analysis, and an extension of the Schwartz-Zippel lemma to compositions of
polynomials.Comment: fixed a bug in the proof of claim 5.6 (now lemma 5.5
List-decoding reed-muller codes over small fields
We present the first local list-decoding algorithm for the rth order Reed-Muller code RM(r,m) over F2 for r ≥ 2. Given an oracle for a received word R: Fm2 → F2, our random-ized local list-decoding algorithm produces a list containing all degree r polynomials within relative distance (2−r − ε) from R for any ε> 0 in time poly(mr, ε−r). The list size could be exponential in m at radius 2−r, so our bound is op-timal in the local setting. Since RM(r,m) has relative dis-tance 2−r, our algorithm beats the Johnson bound for r ≥ 2. In the setting where we are allowed running-time polyno-mial in the block-length, we show that list-decoding is pos-sible up to even larger radii, beyond the minimum distance. We give a deterministic list-decoder that works at error rate below J(21−r), where J(δ) denotes the Johnson radius for minimum distance δ. This shows that RM(2,m) codes are list-decodable up to radius η for any constant η < 1 2 in time polynomial in the block-length. Over small fields Fq, we present list-decoding algorithms in both the global and local settings that work up to the list-decoding radius. We conjecture that the list-decoding radius approaches the minimum distance (like over F2), and prove this holds true when the degree is divisible by q − 1
List decoding group homomorphisms between supersolvable groups
We show that the set of homomorphisms between two supersolvable groups can be
locally list decoded up to the minimum distance of the code, extending the
results of Dinur et al who studied the case where the groups are abelian.
Moreover, when specialized to the abelian case, our proof is more streamlined
and gives a better constant in the exponent of the list size. The constant is
improved from about 3.5 million to 105.Comment: 11 page
Bias vs structure of polynomials in large fields, and applications in effective algebraic geometry and coding theory
Let be a polynomial of degree in variables over a finite field
. The polynomial is said to be unbiased if the distribution of
for a uniform input is close to the uniform
distribution over , and is called biased otherwise. The polynomial
is said to have low rank if it can be expressed as a composition of a few lower
degree polynomials. Green and Tao [Contrib. Discrete Math 2009] and Kaufman and
Lovett [FOCS 2008] showed that bias implies low rank for fixed degree
polynomials over fixed prime fields. This lies at the heart of many tools in
higher order Fourier analysis. In this work, we extend this result to all prime
fields (of size possibly growing with ). We also provide a generalization to
nonprime fields in the large characteristic case. However, we state all our
applications in the prime field setting for the sake of simplicity of
presentation.
As an immediate application, we obtain improved bounds for a suite of
problems in effective algebraic geometry, including Hilbert nullstellensatz,
radical membership and counting rational points in low degree varieties.
Using the above generalization to large fields as a starting point, we are
also able to settle the list decoding radius of fixed degree Reed-Muller codes
over growing fields. The case of fixed size fields was solved by Bhowmick and
Lovett [STOC 2015], which resolved a conjecture of Gopalan-Klivans-Zuckerman
[STOC 2008]. Here, we show that the list decoding radius is equal the minimum
distance of the code for all fixed degrees, even when the field size is
possibly growing with
Group homomorphisms as error correcting codes
We investigate the minimum distance of the error correcting code formed by
the homomorphisms between two finite groups and . We prove some general
structural results on how the distance behaves with respect to natural group
operations, such as passing to subgroups and quotients, and taking products.
Our main result is a general formula for the distance when is solvable or
is nilpotent, in terms of the normal subgroup structure of as well as
the prime divisors of and . In particular, we show that in the above
case, the distance is independent of the subgroup structure of . We
complement this by showing that, in general, the distance depends on the
subgroup structure .Comment: 13 page
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
Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex
Reed-Muller codes are among the most important classes of locally correctable
codes. Currently local decoding of Reed-Muller codes is based on decoding on
lines or quadratic curves to recover one single coordinate. To recover multiple
coordinates simultaneously, the naive way is to repeat the local decoding for
recovery of a single coordinate. This decoding algorithm might be more
expensive, i.e., require higher query complexity. In this paper, we focus on
Reed-Muller codes with usual parameter regime, namely, the total degree of
evaluation polynomials is , where is the code alphabet size
(in fact, can be as big as in our setting). By introducing a novel
variation of codex, i.e., interleaved codex (the concept of codex has been used
for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover
arbitrarily large number of coordinates of a Reed-Muller code
simultaneously at the cost of querying coordinates. It turns out that
our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that
accessing locations is in fact cheaper than repeating the procedure for
accessing a single location for times. Our estimation of success error
probability is based on error probability bound for -wise linearly
independent variables given in \cite{BR94}