4 research outputs found
On the similarities between generalized rank and Hamming weights and their applications to network coding
Rank weights and generalized rank weights have been proven to characterize
error and erasure correction, and information leakage in linear network coding,
in the same way as Hamming weights and generalized Hamming weights describe
classical error and erasure correction, and information leakage in wire-tap
channels of type II and code-based secret sharing. Although many similarities
between both cases have been established and proven in the literature, many
other known results in the Hamming case, such as bounds or characterizations of
weight-preserving maps, have not been translated to the rank case yet, or in
some cases have been proven after developing a different machinery. The aim of
this paper is to further relate both weights and generalized weights, show that
the results and proofs in both cases are usually essentially the same, and see
the significance of these similarities in network coding. Some of the new
results in the rank case also have new consequences in the Hamming case
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