666 research outputs found
Bounds on List Decoding of Rank-Metric Codes
So far, there is no polynomial-time list decoding algorithm (beyond half the
minimum distance) for Gabidulin codes. These codes can be seen as the
rank-metric equivalent of Reed--Solomon codes. In this paper, we provide bounds
on the list size of rank-metric codes in order to understand whether
polynomial-time list decoding is possible or whether it works only with
exponential time complexity. Three bounds on the list size are proven. The
first one is a lower exponential bound for Gabidulin codes and shows that for
these codes no polynomial-time list decoding beyond the Johnson radius exists.
Second, an exponential upper bound is derived, which holds for any rank-metric
code of length and minimum rank distance . The third bound proves that
there exists a rank-metric code over \Fqm of length such that the
list size is exponential in the length for any radius greater than half the
minimum rank distance. This implies that there cannot exist a polynomial upper
bound depending only on and similar to the Johnson bound in Hamming
metric. All three rank-metric bounds reveal significant differences to bounds
for codes in Hamming metric.Comment: 10 pages, 2 figures, submitted to IEEE Transactions on Information
Theory, short version presented at ISIT 201
Interpolation-Based Decoding of Folded Variants of Linearized and Skew Reed-Solomon Codes
The sum-rank metric is a hybrid between the Hamming metric and the rank
metric and suitable for error correction in multishot network coding and
distributed storage as well as for the design of quantum-resistant
cryptosystems. In this work, we consider the construction and decoding of
folded linearized Reed-Solomon (FLRS) codes, which are shown to be maximum
sum-rank distance (MSRD) for appropriate parameter choices. We derive an
efficient interpolation-based decoding algorithm for FLRS codes that can be
used as a list decoder or as a probabilistic unique decoder. The proposed
decoding scheme can correct sum-rank errors beyond the unique decoding radius
with a computational complexity that is quadratic in the length of the unfolded
code. We show how the error-correction capability can be optimized for
high-rate codes by an alternative choice of interpolation points. We derive a
heuristic upper bound on the decoding failure probability of the probabilistic
unique decoder and verify its tightness by Monte Carlo simulations. Further, we
study the construction and decoding of folded skew Reed-Solomon codes in the
skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with
different block sizes that come along with an efficient decoding algorithm.Comment: 32 pages, 3 figures, accepted at Designs, Codes and Cryptograph
Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants
We construct s-interleaved linearized Reed-Solomon (ILRS) codes and variants
and propose efficient decoding schemes that can correct errors beyond the
unique decoding radius in the sum-rank, sum-subspace and skew metric. The
proposed interpolation-based scheme for ILRS codes can be used as a list
decoder or as a probabilistic unique decoder that corrects errors of sum-rank
up to , where s is the interleaving order, n the
length and k the dimension of the code. Upper bounds on the list size and the
decoding failure probability are given where the latter is based on a novel
Loidreau-Overbeck-like decoder for ILRS codes. The results are extended to
decoding of lifted interleaved linearized Reed-Solomon (LILRS) codes in the
sum-subspace metric and interleaved skew Reed-Solomon (ISRS) codes in the skew
metric. We generalize fast minimal approximant basis interpolation techniques
to obtain efficient decoding schemes for ILRS codes (and variants) with
subquadratic complexity in the code length. Up to our knowledge, the presented
decoding schemes are the first being able to correct errors beyond the unique
decoding region in the sum-rank, sum-subspace and skew metric. The results for
the proposed decoding schemes are validated via Monte Carlo simulations.Comment: submitted to IEEE Transactions on Information Theory, 57 pages, 10
figure
Decoding and constructions of codes in rank and Hamming metric
As coding theory plays an important role in data transmission, decoding algorithms for new families of error correction codes are of great interest. This dissertation is dedicated to the decoding algorithms for new families of maximum rank distance (MRD) codes including additive generalized twisted Gabidulin (AGTG) codes and Trombetti-Zhou (TZ) codes, decoding algorithm for Gabidulin codes beyond half the minimum distance and also encoding and decoding algorithms for some new optimal rank metric codes with restrictions.
We propose an interpolation-based decoding algorithm to decode AGTG codes where the decoding problem is reduced to the problem of solving a projective polynomial equation of the form q(x) = xqu+1 +bx+a = 0 for a,b ∈ Fqm. We investigate the zeros of q(x) when gcd(u,m)=1 and proposed a deterministic algorithm to solve a linearized polynomial equation which has a close connection to the zeros of q(x).
An efficient polynomial-time decoding algorithm is proposed for TZ codes. The interpolation-based decoding approach transforms the decoding problem of TZ codes to the problem of solving a quadratic polynomial equation. Two new communication models are defined and using our models we manage to decode Gabidulin codes beyond half the minimum distance by one unit. Our models also allow us to improve the complexity for decoding GTG and AGTG codes.
Besides working on MRD codes, we also work on restricted optimal rank metric codes including symmetric, alternating and Hermitian rank metric codes. Both encoding and decoding algorithms for these optimal families are proposed. In all the decoding algorithms presented in this thesis, the properties of Dickson matrix and the BM algorithm play crucial roles.
We also touch two problems in Hamming metric. For the first problem, some cryptographic properties of Welch permutation polynomial are investigated and we use these properties to determine the weight distribution of a binary linear codes with few weights. For the second one, we introduce two new subfamilies for maximum weight spectrum codes with respect to their weight distribution and then we investigate their properties.Doktorgradsavhandlin
Integer sequences that are generalized weights of a linear code
Which integer sequences are sequences of generalized weights of a linear
code? In this paper, we answer this question for linear block codes,
rank-metric codes, and more generally for sum-rank metric codes. We do so under
an existence assumption for MDS and MSRD codes. We also prove that the same
integer sequences appear as sequences of greedy weights of linear block codes,
rank-metric codes, and sum-rank metric codes. Finally, we characterize the
integer sequences which appear as sequences of relative generalized weights
(respectively, relative greedy weights) of linear block codes.Comment: 19 page
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