Papers by Antonia Wachter
Partial) Unit Memory ((P)UM) codes provide a powerful possibility to construct convolutional code... more Partial) Unit Memory ((P)UM) codes provide a powerful possibility to construct convolutional codes based on block codes in order to achieve a high decoding performance. In this contribution, a construction based on Gabidulin codes is considered. This construction requires a modified rank metric, the so-called sum rank metric. For the sum rank metric, the free rank distance, the extended row rank distance and its slope are defined analogous to the extended row distance in Hamming metric. Upper bounds for the free rank distance and the slope of (P)UM codes in the sum rank metric are derived and an explicit construction of (P)UM codes based on Gabidulin codes is given, achieving the upper bound for the free rank distance.
The Schmidt–Sidorenko–Bossert scheme extends a low-rate Reed–Solomon code to an Interleaved Reed–... more The Schmidt–Sidorenko–Bossert scheme extends a low-rate Reed–Solomon code to an Interleaved Reed–Solomon code and achieves the decoding radius of Sudan’s original list decoding algorithm while the decoding result remains unambiguous. We adapt this result to the case of Generalized Reed–Solomon codes and calculate the parameters of the corresponding Interleaved Generalized Reed–Solomon code. Furthermore, the failure probability is derived.
2010 IEEE International Symposium on Information Theory, 2010
We present and prove the correctness of an efficient algorithm that provides a basis for all solu... more We present and prove the correctness of an efficient algorithm that provides a basis for all solutions of a key equation in order to decode Gabidulin (G-) codes up to a given radius τ. This algorithm is based on a symbolic equivalent of the Euclidean Algorithm (EA) and can be applied for decoding of G-codes beyond half the minimum rank distance. If the key equation has a unique solution, our algorithm reduces to Gabidulin's decoding algorithm up to half the minimum distance. If the solution is not unique, we provide a basis for all solutions of the key equation. Our algorithm has time complexity O(τ 2) and is a generalization of the modified EA by Bossert and Bezzateev for Reed-Solomon codes.
A fast linearized Euclidean algorithm for decoding Gabidulin codes
We present a fast equivalent of the Extended Euclidean Algorithm for linearized polynomials (LEEA... more We present a fast equivalent of the Extended Euclidean Algorithm for linearized polynomials (LEEA). Linearized polynomials are used to define rank metric codes, e.g. Gabidulin codes. The proposed fast LEEA accelerates decoding of Gabidulin codes since it solves the key equation with sub-quadratic complexity. In addition, we give a fast algorithm for calculating the symbolic product.
A new bound on the distance of binary cyclic codes is proposed. The approach is based on the repr... more A new bound on the distance of binary cyclic codes is proposed. The approach is based on the representation of a subset of the roots of the generator polynomial by a rational function. A new bound on the minimum distance is proven and several classes of binary cyclic codes are identified. For some classes of codes, this bound is better than the known bounds (e.g. BCH or Hartmann-Tzeng bound). Furthermore, a quadratic-time decoding algorithm up to this new bound is developed.
A method for soft-decision decoding of Reed-Solomon codes based on the extended Euclidean algorithm
We present an algorithm for decoding Reed-Solomon codes beyond half the minimum distance by using... more We present an algorithm for decoding Reed-Solomon codes beyond half the minimum distance by using reliability information which is based on the extended Euclidean algorithm. The algorithm constitutes a Generalized Minimum Distance decoder since the reliability information is used to declare erasures in certain positions in the received word. We describe two methods to reduce the decoding complexity of this decoder.
Problems of Information Transmission, 2011
Partial) Unit Memory ((P)UM) codes provide a powerful possibility to construct convolutional code... more Partial) Unit Memory ((P)UM) codes provide a powerful possibility to construct convolutional codes based on block codes in order to achieve a high decoding performance. In this contribution, a construction based on Gabidulin codes is considered. This construction requires a modified rank metric, the so-called sum rank metric. For the sum rank metric, the free rank distance, the extended row rank distance and its slope are defined analogous to the extended row distance in Hamming metric. Upper bounds for the free rank distance and the slope of (P)UM codes in the sum rank metric are derived and an explicit construction of (P)UM codes based on Gabidulin codes is given, achieving the upper bound for the free rank distance.
IEEE Transactions on Information Theory, 2012
A new lower bound on the minimum distance of qary cyclic codes is proposed. This bound improves u... more A new lower bound on the minimum distance of qary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean Algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula.
Designs, Codes and Cryptography, 2012
A new lower bound on the minimum distance of qary cyclic codes is proposed. This bound improves u... more A new lower bound on the minimum distance of qary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean Algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula.
Advances in Mathematics of Communications, 2011
Feng and Tzeng's generalization of the Extended Euclidean Algorithm synthesizes the shortest-leng... more Feng and Tzeng's generalization of the Extended Euclidean Algorithm synthesizes the shortest-length linear feedback shift-register for s ≥ 1 sequences, where each sequence has the same length n. In this contribution, it is shown that Feng and Tzeng's algorithm which solves this multi-sequence shift-register problem has time complexity O(sn 2). An acceleration based on the Divide and Conquer strategy is proposed and it is proven that subquadratic time complexity is achieved.
Uploads
Papers by Antonia Wachter