13 research outputs found
Solving Shift Register Problems over Skew Polynomial Rings using Module Minimisation
For many algebraic codes the main part of decoding can be reduced to a shift
register synthesis problem. In this paper we present an approach for solving
generalised shift register problems over skew polynomial rings which occur in
error and erasure decoding of -Interleaved Gabidulin codes. The algorithm
is based on module minimisation and has time complexity where
measures the size of the input problem.Comment: 10 pages, submitted to WCC 201
Generic interpolation polynomial for list decoding
AbstractWe extend results of K. Lee and M.E. OʼSullivan by showing how to use Gröbner bases to find the interpolation polynomial for list decoding a one-point AG code C=CL(rP,D) on any curve X, where P is an Fq-rational point on X and D=P1+P2+⋯+Pn is the sum of other Fq-rational points on X. We then define the generic interpolation polynomial for list decoding such a code. The generic interpolation polynomial should specialize to the interpolation polynomial for most received strings. We give an example of a family of Reed–Solomon 1-error correcting codes for which a single error can be decoded by a very simple process involving substituting into the generic interpolation polynomial
Application of Module to Coding Theory: A Systematic Literature Review
A systematic literature review is a research process that identifies,
evaluates, and interprets all relevant study findings connected to specific
research questions, topics, or phenomena of interest. In this work, a thorough
review of the literature on the issue of the link between module structure and
coding theory was done. A literature search yielded 470 articles from the
Google Scholar, Dimensions, and Science Direct databases. After further article
selection process, 14 articles were chosen to be studied in further depth. The
items retrieved were from the previous ten years, from 2012 to 2022. The PRISMA
analytical approach and bibliometric analysis were employed in this
investigation. A more detailed description of the PRISMA technique and the
significance of the bibliometric analysis is provided. The findings of this
study are presented in the form of brief summaries of the 14 articles and
research recommendations. At the end of the study, recommendations for future
development of the code structure utilized in the articles that are further
investigated are made
List Decoding Algorithm based on Voting in Groebner Bases for General One-Point AG Codes
We generalize the unique decoding algorithm for one-point AG codes over the
Miura-Kamiya Cab curves proposed by Lee, Bras-Amor\'os and O'Sullivan (2012) to
general one-point AG codes, without any assumption. We also extend their unique
decoding algorithm to list decoding, modify it so that it can be used with the
Feng-Rao improved code construction, prove equality between its error
correcting capability and half the minimum distance lower bound by Andersen and
Geil (2008) that has not been done in the original proposal except for
one-point Hermitian codes, remove the unnecessary computational steps so that
it can run faster, and analyze its computational complexity in terms of
multiplications and divisions in the finite field. As a unique decoding
algorithm, the proposed one is empirically and theoretically as fast as the BMS
algorithm for one-point Hermitian codes. As a list decoding algorithm,
extensive experiments suggest that it can be much faster for many moderate
size/usual inputs than the algorithm by Beelen and Brander (2010). It should be
noted that as a list decoding algorithm the proposed method seems to have
exponential worst-case computational complexity while the previous proposals
(Beelen and Brander, 2010; Guruswami and Sudan, 1999) have polynomial ones, and
that the proposed method is expected to be slower than the previous proposals
for very large/special inputs.Comment: Accepted for publication in J. Symbolic Computation. LaTeX2e
article.cls, 42 pages, 4 tables, no figures. Ver. 6 added an illustrative
example of the algorithm executio
Row Reduction Applied to Decoding of Rank Metric and Subspace Codes
We show that decoding of -Interleaved Gabidulin codes, as well as
list- decoding of Mahdavifar--Vardy codes can be performed by row
reducing skew polynomial matrices. Inspired by row reduction of \F[x]
matrices, we develop a general and flexible approach of transforming matrices
over skew polynomial rings into a certain reduced form. We apply this to solve
generalised shift register problems over skew polynomial rings which occur in
decoding -Interleaved Gabidulin codes. We obtain an algorithm with
complexity where measures the size of the input problem
and is proportional to the code length in the case of decoding. Further, we
show how to perform the interpolation step of list--decoding
Mahdavifar--Vardy codes in complexity , where is the number of
interpolation constraints.Comment: Accepted for Designs, Codes and Cryptograph