[Show abstract][Hide abstract] ABSTRACT: (Communicated by Michael O’Sullivan) Abstract. 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. 1.
Preview · Article · Nov 2011 · Advances in Mathematics of Communications
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: A new lower bound on the minimum distance of q-ary 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.
Full-text · Article · May 2011 · IEEE Transactions on Information Theory
[Show abstract][Hide abstract] ABSTRACT: (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.
Full-text · Article · Feb 2011 · Problems of Information Transmission
[Show abstract][Hide abstract] ABSTRACT: 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 tau. 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(tau^2) and is a generalization of the modified EA by Bossert and Bezzateev for Reed-Solomon codes. Comment: accepted for ISIT 2010, Austin, TX, USA
[Show abstract][Hide abstract] ABSTRACT: 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.