Decoding and constructions of codes in rank and Hamming metric
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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.
Består avPaper 1: Wrya K. Kadir, and Chunlei Li. "On decoding additive generalized twisted Gabidulin codes." Cryptography and Communications (2020): 987-1009. The article is available at: https://hdl.handle.net/11250/2764598
Paper 2: Wrya K. Kadir, Chunlei Li, and Ferdinando Zullo, "On interpolation-based decoding of a class of maximum rank distance codes." 2021 IEEE International Symposium on Information Theory (ISIT), (2021): 31-36. The accepted version is available in the thesis file. The published version is available at: https://doi.org/10.1109/ISIT45174.2021.9517980
Paper 3: Wrya K. Kadir "New Communication Models and Decoding of Maximum Rank Distance Codes. " 2021 XVII International Symposium "Problems of Redundancy in Information and Control Systems" (REDUNDANCY), (2021): 125-130. The accepted version is available in the thesis file. The published version is available at: https://doi.org/10.1109/REDUNDANCY52534.2021.9606451
Paper 4: Wrya K. Kadir, Chunlei Li, and Ferdinando Zullo. "Encoding and decoding of several optimal rank metric codes." Cryptography and Communications (2022): 1-20. The article is available in the thesis file. The article is also available at: https://doi.org/10.1007/s12095-022-00578-3
Paper 5: YiboWang , Wrya K. Kadir, Chunlei Li, and Yongbo Xia. "On cryptographic properties of the Welch permutation and a related conjecture." International Conference on Sequences and Their Applications (SETA) Russia, Saint-Petersburg, 2020. The article is available in the thesis file.
Paper 6: Alessio Meneghetti, and Wrya K. Kadir. "Characterisation of the parameters of maximum weight spectrum codes according to their spread." International Conference on Sequences and Their Applications (SETA) Russia, Saint-Petersburg, 2020. The article is available in the thesis file.