4,280 research outputs found
On Algebraic Decoding of -ary Reed-Muller and Product-Reed-Solomon Codes
We consider a list decoding algorithm recently proposed by Pellikaan-Wu
\cite{PW2005} for -ary Reed-Muller codes of
length when . A simple and easily accessible
correctness proof is given which shows that this algorithm achieves a relative
error-correction radius of . This is
an improvement over the proof using one-point Algebraic-Geometric codes given
in \cite{PW2005}. The described algorithm can be adapted to decode
Product-Reed-Solomon codes.
We then propose a new low complexity recursive algebraic decoding algorithm
for Reed-Muller and Product-Reed-Solomon codes. Our algorithm achieves a
relative error correction radius of . This technique is then proved to outperform the Pellikaan-Wu
method in both complexity and error correction radius over a wide range of code
rates.Comment: 5 pages, 5 figures, to be presented at 2007 IEEE International
Symposium on Information Theory, Nice, France (ISIT 2007
AG codes achieve list decoding capacity over contant-sized fields
The recently-emerging field of higher order MDS codes has sought to unify a
number of concepts in coding theory. Such areas captured by higher order MDS
codes include maximally recoverable (MR) tensor codes, codes with optimal
list-decoding guarantees, and codes with constrained generator matrices (as in
the GM-MDS theorem).
By proving these equivalences, Brakensiek-Gopi-Makam showed the existence of
optimally list-decodable Reed-Solomon codes over exponential sized fields.
Building on this, recent breakthroughs by Guo-Zhang and Alrabiah-Guruswami-Li
have shown that randomly punctured Reed-Solomon codes achieve list-decoding
capacity (which is a relaxation of optimal list-decodability) over linear size
fields. We extend these works by developing a formal theory of relaxed higher
order MDS codes. In particular, we show that there are two inequivalent
relaxations which we call lower and upper relaxations. The lower relaxation is
equivalent to relaxed optimal list-decodable codes and the upper relaxation is
equivalent to relaxed MR tensor codes with a single parity check per column.
We then generalize the techniques of GZ and AGL to show that both these
relaxations can be constructed over constant size fields by randomly puncturing
suitable algebraic-geometric codes. For this, we crucially use the generalized
GM-MDS theorem for polynomial codes recently proved by Brakensiek-Dhar-Gopi. We
obtain the following corollaries from our main result. First, randomly
punctured AG codes of rate achieve list-decoding capacity with list size
and field size . Prior to this work, AG
codes were not even known to achieve list-decoding capacity. Second, by
randomly puncturing AG codes, we can construct relaxed MR tensor codes with a
single parity check per column over constant-sized fields, whereas
(non-relaxed) MR tensor codes require exponential field size.Comment: 38 page
Generalized GM-MDS: Polynomial Codes are Higher Order MDS
The GM-MDS theorem, conjectured by Dau-Song-Dong-Yuen and proved by Lovett
and Yildiz-Hassibi, shows that the generator matrices of Reed-Solomon codes can
attain every possible configuration of zeros for an MDS code. The recently
emerging theory of higher order MDS codes has connected the GM-MDS theorem to
other important properties of Reed-Solomon codes, including showing that
Reed-Solomon codes can achieve list decoding capacity, even over fields of size
linear in the message length.
A few works have extended the GM-MDS theorem to other families of codes,
including Gabidulin and skew polynomial codes. In this paper, we generalize all
these previous results by showing that the GM-MDS theorem applies to any
\emph{polynomial code}, i.e., a code where the columns of the generator matrix
are obtained by evaluating linearly independent polynomials at different
points. We also show that the GM-MDS theorem applies to dual codes of such
polynomial codes, which is non-trivial since the dual of a polynomial code may
not be a polynomial code. More generally, we show that GM-MDS theorem also
holds for algebraic codes (and their duals) where columns of the generator
matrix are chosen to be points on some irreducible variety which is not
contained in a hyperplane through the origin. Our generalization has
applications to constructing capacity-achieving list-decodable codes as shown
in a follow-up work by Brakensiek-Dhar-Gopi-Zhang, where it is proved that
randomly punctured algebraic-geometric (AG) codes achieve list-decoding
capacity over constant-sized fields.Comment: 34 page
Evaluation codes defined by finite families of plane valuations at infinity
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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))))
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