207 research outputs found
List and Unique Error-Erasure Decoding of Interleaved Gabidulin Codes with Interpolation Techniques
A new interpolation-based decoding principle for interleaved Gabidulin codes
is presented. The approach consists of two steps: First, a multi-variate
linearized polynomial is constructed which interpolates the coefficients of the
received word and second, the roots of this polynomial have to be found. Due to
the specific structure of the interpolation polynomial, both steps
(interpolation and root-finding) can be accomplished by solving a linear system
of equations. This decoding principle can be applied as a list decoding
algorithm (where the list size is not necessarily bounded polynomially) as well
as an efficient probabilistic unique decoding algorithm. For the unique
decoder, we show a connection to known unique decoding approaches and give an
upper bound on the failure probability. Finally, we generalize our approach to
incorporate not only errors, but also row and column erasures.Comment: accepted for Designs, Codes and Cryptography; presented in part at
WCC 2013, Bergen, Norwa
Construction of Quasi-Cyclic Product Codes
Linear quasi-cyclic product codes over finite fields are investigated. Given
the generating set in the form of a reduced Gr{\"o}bner basis of a quasi-cyclic
component code and the generator polynomial of a second cyclic component code,
an explicit expression of the basis of the generating set of the quasi-cyclic
product code is given. Furthermore, the reduced Gr{\"o}bner basis of a
one-level quasi-cyclic product code is derived.Comment: 10th International ITG Conference on Systems, Communications and
Coding (SCC), Feb 2015, Hamburg, German
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
Optimal Linear and Cyclic Locally Repairable Codes over Small Fields
We consider locally repairable codes over small fields and propose
constructions of optimal cyclic and linear codes in terms of the dimension for
a given distance and length. Four new constructions of optimal linear codes
over small fields with locality properties are developed. The first two
approaches give binary cyclic codes with locality two. While the first
construction has availability one, the second binary code is characterized by
multiple available repair sets based on a binary Simplex code. The third
approach extends the first one to q-ary cyclic codes including (binary)
extension fields, where the locality property is determined by the properties
of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear
codes with locality greater than two are obtained by the fourth construction.Comment: IEEE Information Theory Workshop (ITW) 2015, Apr 2015, Jerusalem,
Israe
Optimal Binary Locally Repairable Codes via Anticodes
This paper presents a construction for several families of optimal binary
locally repairable codes (LRCs) with small locality (2 and 3). This
construction is based on various anticodes. It provides binary LRCs which
attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal
with respect to the Griesmer bound
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
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