7 research outputs found
Multi-Trial Guruswami–Sudan Decoding for Generalised Reed–Solomon Codes
An iterated refinement procedure for the Guruswami--Sudan list decoding
algorithm for Generalised Reed--Solomon codes based on Alekhnovich's module
minimisation is proposed. The method is parametrisable and allows variants of
the usual list decoding approach. In particular, finding the list of
\emph{closest} codewords within an intermediate radius can be performed with
improved average-case complexity while retaining the worst-case complexity.Comment: WCC 2013 International Workshop on Coding and Cryptography (2013
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
On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes
We derive the Wu list-decoding algorithm for Generalised Reed-Solomon (GRS)
codes by using Gr\"obner bases over modules and the Euclidean algorithm (EA) as
the initial algorithm instead of the Berlekamp-Massey algorithm (BMA). We
present a novel method for constructing the interpolation polynomial fast. We
give a new application of the Wu list decoder by decoding irreducible binary
Goppa codes up to the binary Johnson radius. Finally, we point out a connection
between the governing equations of the Wu algorithm and the Guruswami-Sudan
algorithm (GSA), immediately leading to equality in the decoding range and a
duality in the choice of parameters needed for decoding, both in the case of
GRS codes and in the case of Goppa codes.Comment: To appear in IEEE Transactions of Information Theor
Generalised Multi-sequence Shift-Register synthesis using module minimisation
We show how to solve a generalised version of the Multi-sequence Linear
Feedback Shift-Register (MLFSR) problem using minimisation of free modules over
. We show how two existing algorithms for minimising such modules
run particularly fast on these instances. Furthermore, we show how one of them
can be made even faster for our use. With our modeling of the problem,
classical algebraic results tremendously simplify arguing about the algorithms.
For the non-generalised MLFSR, these algorithms are as fast as what is
currently known. We then use our generalised MLFSR to give a new fast decoding
algorithm for Reed Solomon codes.Comment: Version with full proofs. Fixed some typos in algorithm from last
version. Presented at ISIT 201