6,068 research outputs found
Subquadratic time encodable codes beating the Gilbert-Varshamov bound
We construct explicit algebraic geometry codes built from the
Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for
alphabet sizes at least 192. Messages are identied with functions in certain
Riemann-Roch spaces associated with divisors supported on multiple places.
Encoding amounts to evaluating these functions at degree one places. By
exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we
devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and
1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list)
decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent.
If \omega = 2, as widely believed, the encoding and decoding runtimes are
respectively nearly linear and nearly quadratic. Prior to this work, encoding
(resp. decoding) time of code families beating the Gilbert-Varshamov bound were
quadratic (resp. cubic) or worse
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}
Asymptotically good binary linear codes with asymptotically good self-intersection spans
If C is a binary linear code, let C^2 be the linear code spanned by
intersections of pairs of codewords of C. We construct an asymptotically good
family of binary linear codes such that, for C ranging in this family, the C^2
also form an asymptotically good family. For this we use algebraic-geometry
codes, concatenation, and a fair amount of bilinear algebra.
More precisely, the two main ingredients used in our construction are, first,
a description of the symmetric square of an odd degree extension field in terms
only of field operations of small degree, and second, a recent result of
Garcia-Stichtenoth-Bassa-Beelen on the number of points of curves on such an
odd degree extension field.Comment: 18 pages; v2->v3: expanded introduction and bibliography + various
minor change
Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs
A novel technique, based on the pseudo-random properties of certain graphs known as expanders, is used to obtain novel simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling, and then regrouping the code coordinates. For any fixed (small) rate, and for a sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF(2)) as well. Although these concatenated codes lie below the Zyablov bound, they are still superior to previously known explicit constructions in the zero-rate neighborhood
Incidence structures from the blown-up plane and LDPC codes
In this article, new regular incidence structures are presented. They arise
from sets of conics in the affine plane blown-up at its rational points. The
LDPC codes given by these incidence matrices are studied. These sparse
incidence matrices turn out to be redundant, which means that their number of
rows exceeds their rank. Such a feature is absent from random LDPC codes and is
in general interesting for the efficiency of iterative decoding. The
performance of some codes under iterative decoding is tested. Some of them turn
out to perform better than regular Gallager codes having similar rate and row
weight.Comment: 31 pages, 10 figure
On products and powers of linear codes under componentwise multiplication
In this text we develop the formalism of products and powers of linear codes
under componentwise multiplication. As an expanded version of the author's talk
at AGCT-14, focus is put mostly on basic properties and descriptive statements
that could otherwise probably not fit in a regular research paper. On the other
hand, more advanced results and applications are only quickly mentioned with
references to the literature. We also point out a few open problems.
Our presentation alternates between two points of view, which the theory
intertwines in an essential way: that of combinatorial coding, and that of
algebraic geometry.
In appendices that can be read independently, we investigate topics in
multilinear algebra over finite fields, notably we establish a criterion for a
symmetric multilinear map to admit a symmetric algorithm, or equivalently, for
a symmetric tensor to decompose as a sum of elementary symmetric tensors.Comment: 75 pages; expanded version of a talk at AGCT-14 (Luminy), to appear
in vol. 637 of Contemporary Math., AMS, Apr. 2015; v3: minor typos corrected
in the final "open questions" sectio
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