Iterative List-Decoding of Gabidulin Codes via Gr\"obner Based Interpolation

We show how Gabidulin codes can be list decoded by using an iterative parametrization approach. For a given received word, our decoding algorithm processes its entries one by one, constructing four polynomials at each step. This then yields a parametrization of interpolating solutions for the data so far. From the final result a list of all codewords that are closest to the received word with respect to the rank metric is obtained.Comment: Submitted to IEEE Information Theory Workshop 2014 in Hobart, Australi

List-Decoding Gabidulin Codes via Interpolation and the Euclidean Algorithm

We show how Gabidulin codes can be list decoded by using a parametrization approach. For this we consider a certain module in the ring of linearized polynomials and find a minimal basis for this module using the Euclidean algorithm with respect to composition of polynomials. For a given received word, our decoding algorithm computes a list of all codewords that are closest to the received word with respect to the rank metric.Comment: Submitted to ISITA 2014, IEICE copyright upon acceptanc

List Decoding Algorithms based on Groebner Bases for General One-Point AG Codes

We generalize the list decoding algorithm for Hermitian codes proposed by Lee and O'Sullivan based on Gr\"obner bases to general one-point AG codes, under an assumption weaker than one used by Beelen and Brander. By using the same principle, we also generalize the unique decoding algorithm for one-point AG codes over the Miura-Kamiya $C_{ab}$ curves proposed by Lee, Bras-Amor\'os and O'Sullivan to general one-point AG codes, without any assumption. Finally we extend the latter 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 that has not been done in the original proposal, and remove the unnecessary computational steps so that it can run faster.Comment: IEEEtran.cls, 5 pages, no figure. To appear in Proc. 2012 IEEE International Symposium on Information Theory, July 1-6, 2012, Boston, MA, USA. Version 4 corrected wrong description of the work by Lee, Bras-Amor\'os and O'Sullivan, and added four reference

An iterative algorithm for parametrization of shortest length shift registers over finite rings

The construction of shortest feedback shift registers for a finite sequence S_1,...,S_N is considered over the finite ring Z_{p^r}. A novel algorithm is presented that yields a parametrization of all shortest feedback shift registers for the sequence of numbers S_1,...,S_N, thus solving an open problem in the literature. The algorithm iteratively processes each number, starting with S_1, and constructs at each step a particular type of minimal Gr\"obner basis. The construction involves a simple update rule at each step which leads to computational efficiency. It is shown that the algorithm simultaneously computes a similar parametrization for the reciprocal sequence S_N,...,S_1.Comment: Submitte

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

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

Метод резервування та відновлення пакетів даних в глобальних мережах та програмні засоби його моделювання

Ціллю представлених в дипломному проекті досліджень є забезпечення надійності зберігання даних користувачів на віддалених серверах на основі хмарних технологій за рахунок резервування інформації та її відновлення при втраті. Формування резервних блоків пропонується здійснювати на основі лінійних перетворень, що дозволяє спростити та прискорити процес реконструювання втрачених даних. Для підвищення ефективності запропоновані та досліджені математичні моделі, які дозволяють використовувати мінімальну кількість резервних блоків для відновлення фіксованої кількості втрачених інформаційних блоків. Сформульовані умови гарантування відновлення всіх інформаційних блоків, за умови, що загальна кількість втрачених блоків не перевищує трьох. Розроблено організацію резервування даних із використанням лінійних перетворень за допомогою мінімальної кількості резервних блоків за умови, що кількість втрачених блоків (основних і резервних) не перевищує трьох. Результати досліджень можуть бути використані для підвищення надійності довготривалого зберігання даних користувачів в хмарних сховищах.The goal of presented by diploma project research is to point out the problem of ensuring the reliability of user data storage on remote servers based on cloud technologies by backing up information and recovering it in case of loss. The formation of backup blocks is proposed to be carried out on the basis of linear transformations, which simplifies and accelerates the process of reconstruction of lost data. To increase the efficiency mathematical models are proposed and researched, which allow to use the minimum number of backup blocks to recover a fixed number of lost information blocks. The conditions for guaranteeing the recovery of all information blocks are formulated, provided that the total number of lost blocks does not exceed three. The organization of data redundancy with the use of linear transformations with the help of the minimum number of reserve blocks is developed, provided that the number of lost blocks (main and reserve) does not exceed three. The results of research can be used to increase the reliability of long-term storage of user data in cloud storage.Целью выполненных в рамках дипломного проекта исследований является обеспечение надежности хранения данных пользователей на удаленных серверах на основе облачных технологий за счет резервирования информации и ее восстановление при потере. Формирование резервных блоков предлагается осуществлять на основе линейных преобразований, позволяет упростить и ускорить процесс реконструкции утраченных данных. Для повышения эффективности предложены и исследованы математические модели, которые позволяют использовать минимальное количество резервных блоков для восстановления фиксированного количества потерянных информационных блоков. Сформулированные условия обеспечения восстановления всех информационных блоков, при условии, что общее количество потерянных блоков не превышает трех. Разработано организацию резервирования данных с использованием линейных преобразований с помощью минимального количества резервных блоков при условии, что количество утраченных блоков не превышает трех. Результаты исследований могут быть использованы для повышения надежности хранения данных пользователей в облачных хранилищах