933 research outputs found
Decoding of Repeated-Root Cyclic Codes up to New Bounds on Their Minimum Distance
The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its
generalization by Hartmann and Tzeng are lower bounds on the minimum distance
of simple-root cyclic codes. We generalize these two bounds to the case of
repeated-root cyclic codes and present a syndrome-based burst error decoding
algorithm with guaranteed decoding radius based on an associated folded cyclic
code. Furthermore, we present a third technique for bounding the minimum
Hamming distance based on the embedding of a given repeated-root cyclic code
into a repeated-root cyclic product code. A second quadratic-time probabilistic
burst error decoding procedure based on the third bound is outlined. Index
Terms Bound on the minimum distance, burst error, efficient decoding, folded
code, repeated-root cyclic code, repeated-root cyclic product cod
Decoding Cyclic Codes up to a New Bound on the Minimum Distance
A new lower bound on the minimum distance of q-ary cyclic codes is proposed.
This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for
some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are
special cases of our bound. For some classes of codes the bound on the minimum
distance is refined. Furthermore, a quadratic-time decoding algorithm up to
this new bound is developed. The determination of the error locations is based
on the Euclidean Algorithm and a modified Chien search. The error evaluation is
done by solving a generalization of Forney's formula
Numerical cubature using error-correcting codes
We present a construction for improving numerical cubature formulas with
equal weights and a convolution structure, in particular equal-weight product
formulas, using linear error-correcting codes. The construction is most
effective in low degree with extended BCH codes. Using it, we obtain several
sequences of explicit, positive, interior cubature formulas with good
asymptotics for each fixed degree as the dimension . Using a
special quadrature formula for the interval [arXiv:math.PR/0408360], we obtain
an equal-weight -cubature formula on the -cube with O(n^{\floor{t/2}})
points, which is within a constant of the Stroud lower bound. We also obtain
-cubature formulas on the -sphere, -ball, and Gaussian with
points when is odd. When is spherically symmetric and
, we obtain points. For each , we also obtain explicit,
positive, interior formulas for the -simplex with points; for
, we obtain O(n) points. These constructions asymptotically improve the
non-constructive Tchakaloff bound.
Some related results were recently found independently by Victoir, who also
noted that the basic construction more directly uses orthogonal arrays.Comment: Dedicated to Wlodzimierz and Krystyna Kuperberg on the occasion of
their 40th anniversary. This version has a major improvement for the n-cub
On Pseudocodewords and Improved Union Bound of Linear Programming Decoding of HDPC Codes
In this paper, we present an improved union bound on the Linear Programming
(LP) decoding performance of the binary linear codes transmitted over an
additive white Gaussian noise channels. The bounding technique is based on the
second-order of Bonferroni-type inequality in probability theory, and it is
minimized by Prim's minimum spanning tree algorithm. The bound calculation
needs the fundamental cone generators of a given parity-check matrix rather
than only their weight spectrum, but involves relatively low computational
complexity. It is targeted to high-density parity-check codes, where the number
of their generators is extremely large and these generators are spread densely
in the Euclidean space. We explore the generator density and make a comparison
between different parity-check matrix representations. That density effects on
the improvement of the proposed bound over the conventional LP union bound. The
paper also presents a complete pseudo-weight distribution of the fundamental
cone generators for the BCH[31,21,5] code
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