46 research outputs found
List Decoding of Locally Repairable Codes
We show that locally repairable codes (LRCs) can be list decoded efficiently
beyond the Johnson radius for a large range of parameters by utilizing the
local error correction capabilities. The new decoding radius is derived and the
asymptotic behavior is analyzed. We give a general list decoding algorithm for
LRCs that achieves this radius along with an explicit realization for a class
of LRCs based on Reed-Solomon codes (Tamo-Barg LRCs). Further, a probabilistic
algorithm for unique decoding of low complexity is given and its success
probability analyzed
List and Probabilistic Unique Decoding of Folded Subspace Codes
A new class of folded subspace codes for noncoherent network coding is
presented. The codes can correct insertions and deletions beyond the unique
decoding radius for any code rate . An efficient interpolation-based
decoding algorithm for this code construction is given which allows to correct
insertions and deletions up to the normalized radius ,
where is the folding parameter and is a decoding parameter. The
algorithm serves as a list decoder or as a probabilistic unique decoder that
outputs a unique solution with high probability. An upper bound on the average
list size of (folded) subspace codes and on the decoding failure probability is
derived. A major benefit of the decoding scheme is that it enables
probabilistic unique decoding up to the list decoding radius.Comment: 6 pages, 1 figure, accepted for ISIT 201
Subspace Designs Based on Algebraic Function Fields
Subspace designs are a (large) collection of high-dimensional subspaces {H_i} of F_q^m such that for any low-dimensional subspace W, only a small number of subspaces from the collection have non-trivial intersection with W; more precisely, the sum of dimensions of W cap H_i is at most some parameter L. The notion was put forth by Guruswami and Xing (STOC\u2713) with applications to list decoding variants of Reed-Solomon and algebraic-geometric codes, and later also used for explicit rank-metric codes with optimal list decoding radius.
Guruswami and Kopparty (FOCS\u2713, Combinatorica\u2716) gave an explicit construction of subspace designs with near-optimal parameters. This construction was based on polynomials and has close connections to folded Reed-Solomon codes, and required large field size (specifically q >= m). Forbes and Guruswami (RANDOM\u2715) used this construction to give explicit constant degree "dimension expanders" over large fields, and noted that subspace designs are a powerful tool in linear-algebraic pseudorandomness.
Here, we construct subspace designs over any field, at the expense of a modest worsening of the bound on total intersection dimension. Our approach is based on a (non-trivial) extension of the polynomial-based construction to algebraic function fields, and instantiating the approach with cyclotomic function fields. Plugging in our new subspace designs in the construction of Forbes and Guruswami yields dimension expanders over F^n for any field F, with logarithmic degree and expansion guarantee for subspaces of dimension Omega(n/(log(log(n))))
Error correction based on partial information
We consider the decoding of linear and array codes from errors when we are
only allowed to download a part of the codeword. More specifically, suppose
that we have encoded data symbols using an code with code length
and dimension During storage, some of the codeword coordinates might
be corrupted by errors. We aim to recover the original data by reading the
corrupted codeword with a limit on the transmitting bandwidth, namely, we can
only download an proportion of the corrupted codeword. For a given
our objective is to design a code and a decoding scheme such that we
can recover the original data from the largest possible number of errors. A
naive scheme is to read coordinates of the codeword. This method
used in conjunction with MDS codes guarantees recovery from any errors. In this paper we show that we can instead read an
proportion from each of the codeword's coordinates. For a
well-designed MDS code, this method can guarantee recovery from errors, which is times more than the naive
method, and is also the maximum number of errors that an code can
correct by downloading only an proportion of the codeword. We present
two families of such optimal constructions and decoding schemes. One is a
Reed-Solomon code with evaluation points in a subfield and the other is based
on Folded Reed-Solomon codes. We further show that both code constructions
attain asymptotically optimal list decoding radius when downloading only a part
of the corrupted codeword. We also construct an ensemble of random codes that
with high probability approaches the upper bound on the number of correctable
errors when the decoder downloads an proportion of the corrupted
codeword.Comment: Extended version of the conference paper in ISIT 201
On the List-Decodability of Random Linear Rank-Metric Codes
The list-decodability of random linear rank-metric codes is shown to match
that of random rank-metric codes. Specifically, an -linear
rank-metric code over of rate is shown to be (with high probability)
list-decodable up to fractional radius with lists of size at
most , where is a constant
depending only on and . This matches the bound for random rank-metric
codes (up to constant factors). The proof adapts the approach of Guruswami,
H\aa stad, Kopparty (STOC 2010), who established a similar result for the
Hamming metric case, to the rank-metric setting