2,297 research outputs found
Combinatorial limitations of average-radius list-decoding
We study certain combinatorial aspects of list-decoding, motivated by the
exponential gap between the known upper bound (of ) and lower
bound (of ) for the list-size needed to decode up to
radius with rate away from capacity, i.e., 1-\h(p)-\gamma (here
and ). Our main result is the following:
We prove that in any binary code of rate
1-\h(p)-\gamma, there must exist a set of
codewords such that the average distance of the
points in from their centroid is at most . In other words,
there must exist codewords with low "average
radius." The standard notion of list-decoding corresponds to working with the
maximum distance of a collection of codewords from a center instead of average
distance. The average-radius form is in itself quite natural and is implied by
the classical Johnson bound.
The remaining results concern the standard notion of list-decoding, and help
clarify the combinatorial landscape of list-decoding:
1. We give a short simple proof, over all fixed alphabets, of the
above-mentioned lower bound. Earlier, this bound
followed from a complicated, more general result of Blinovsky.
2. We show that one {\em cannot} improve the
lower bound via techniques based on identifying the zero-rate regime for list
decoding of constant-weight codes.
3. We show a "reverse connection" showing that constant-weight codes for list
decoding imply general codes for list decoding with higher rate.
4. We give simple second moment based proofs of tight (up to constant
factors) lower bounds on the list-size needed for list decoding random codes
and random linear codes from errors as well as erasures.Comment: 28 pages. Extended abstract in RANDOM 201
It'll probably work out: improved list-decoding through random operations
In this work, we introduce a framework to study the effect of random
operations on the combinatorial list-decodability of a code. The operations we
consider correspond to row and column operations on the matrix obtained from
the code by stacking the codewords together as columns. This captures many
natural transformations on codes, such as puncturing, folding, and taking
subcodes; we show that many such operations can improve the list-decoding
properties of a code. There are two main points to this. First, our goal is to
advance our (combinatorial) understanding of list-decodability, by
understanding what structure (or lack thereof) is necessary to obtain it.
Second, we use our more general results to obtain a few interesting corollaries
for list decoding:
(1) We show the existence of binary codes that are combinatorially
list-decodable from fraction of errors with optimal rate
that can be encoded in linear time.
(2) We show that any code with relative distance, when randomly
folded, is combinatorially list-decodable fraction of errors with
high probability. This formalizes the intuition for why the folding operation
has been successful in obtaining codes with optimal list decoding parameters;
previously, all arguments used algebraic methods and worked only with specific
codes.
(3) We show that any code which is list-decodable with suboptimal list sizes
has many subcodes which have near-optimal list sizes, while retaining the error
correcting capabilities of the original code. This generalizes recent results
where subspace evasive sets have been used to reduce list sizes of codes that
achieve list decoding capacity
Subspace polynomials and list decoding of Reed-Solomon codes
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, February 2007.Includes bibliographical references (p. 29-31).We show combinatorial limitations on efficient list decoding of Reed-Solomon codes beyond the Johnson and Guruswami-Sudan bounds [Joh62, Joh63, GS99]. In particular, we show that for any ... , there exist arbitrarily large fields ... * Existence: there exists a received word ... that agrees with a super-polynomial number of distinct degree K polynomials on ... points each; * Explicit: there exists a polynomial time constructible received word ... that agrees with a super-polynomial number of distinct degree K polynomials, on ... points each. Ill both cases, our results improve upon the previous state of the art, which was , NM/6 for the existence case [JH01], and a ... for the explicit one [GR,05b]. Furthermore, for 6 close to 1 our bound approaches the Guruswami-Sudan bound (which is ... ) and rules out the possibility of extending their efficient RS list decoding algorithm to any significantly larger decoding radius. Our proof method is surprisingly simple. We work with polynomials that vanish on subspaces of an extension field viewed as a vector space over the base field.(cont.) These subspace polynomials are a subclass of linearized polynomials that were studied by Ore [Ore33, Ore34] in the 1930s and by coding theorists. For us their main attraction is their sparsity and abundance of roots. We also complement our negative results by giving a list decoding algorithm for linearized polynomials beyond the Johnson-Guruswami-Sudan bounds.by Swastik Kopparty.S.M
A Lower Bound on List Size for List Decoding
A q-ary error-correcting code C ⊆ {1,2,...,q}n is said to be list decodable to radius ρ with list size L if every Hamming ball of radius ρ contains at most L codewords of C. We prove that in order for a q -ary code to be list-decodable up to radius (1-1/q)(1- ε)n, we must have L = Ω(1/ ε2) . Specifically, we prove that there exists a constant cq > 0 and a function fq such that for small enough ε > 0, if C is list-decodable to radius (1-1/q)(1- ε)n with list size cq/ ε2, then C has at most fq( ε) codewords, independent of n . This result is asymptotically tight (treating q as a constant), since such codes with an exponential (in n ) number of codewords are known for list size L = O(1/ ε2). A result similar to ours is implicit in Blinovsky ( Problems of Information Transmission, 1986) for the binary (q=2) case. Our proof is simpler and works for all alphabet sizes, and provides more intuition for why the lower bound arises.Engineering and Applied Science
- …