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—The key step of syndrome-based decoding of Reed– Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding… (More)

—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… (More)

—The Transmission Control Protocol (TCP) was designed to provide reliable transport services in wired networks. In such networks, packet losses mainly occur due to congestion. Hence, TCP was designed to apply congestion avoidance techniques to cope with packet losses. Nowadays, TCP is also utilized in wireless networks where, besides congestion, numerous… (More)

—We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two approaches give binary cyclic codes with locality two. While the… (More)

—This paper presents a construction for several families of optimal binary locally repairable codes (LRCs) with small locality (2 and 3). This construction is based on various anticodes. It provides binary LRCs which attain the Cadambe– Mazumdar bound. Moreover, most of these codes are optimal with respect to the Griesmer bound. I. INTRODUCTION Locally… (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… (More)

A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann–Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several… (More)

A new probabilistic decoding algorithm for low-rate Interleaved Reed– Solomon (IRS) codes is presented. This approach increases the error correcting capability of IRS codes compared to other known approaches (e.g. joint decoding) with high probability. It is a generalization of well-known decoding approaches and its complexity is quadratic with the length… (More)

—We modify the Euclidean algorithm of Feng and Tzeng to decode Reed–Solomon (RS) codes up to the Sudan radius. The basic steps are the virtual extension to an Interleaved RS code and the reformulation of the multi–sequence shift– register problem of varying length to a multi–sequence problem of equal length. We prove the reformulation and analyze the… (More)