This master thesis has been written for the completion of the author’s study of mathematics and deals with recent developments in coding theory. In particular, descriptions are given of recent algorithms for generating a list of all words form a Reed–Solomon code that are close to a received word. These algorithms may be adapted so that they can take into account the reliability information of ...

We present a randomized algorithm which takes as input n distinct points {(xi, yi)}i=1 from F ×F (where F is a field) and integer parameters t and d and returns a list of all univariate polynomials f over F in the variable x of degree at most d which agree with the given set of points in at least t places (i.e., yi = f(xi) for at least t values of i), provided t = Ω( √ nd). The running time is ...

1 Foreword This text is divided in two parts. The first one, the prerequisites, introduces all the necessary concepts of coding theory. If the reader is familiar with the basics of information and coding theory, he might skip this part and go directly to the second part, which is the core of this thesis. This second part studies an algorithm which improves the decoding of Reed-Solomon codes. Fo...

Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following “curve-fitting” probl...

VLSI Architectures for Modern Error-Correcting Codes CRC Press Book. Reed-Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) codes, and binary. An important class of multipleerror-correcting linear cyclic codes is the class of BCH codes. In fact, BCH code is a generalization of the cyclic Hamming codes. Error correction codes (ECCs) are deployed in digital communication systems to tion 4 recalls...

A list decoding algorithm for matrix-product codes is provided when C1, . . . , Cs are nested linear codes and A is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Further...

For generalized Reed-Solomon codes, it has been proved [6] that the problem of determining if a received word is a deep hole is co-NP-complete. The reduction relies on the fact that the evaluation set of the code can be exponential in the length of the code – a property that practical codes do not usually possess. In this paper, we first presented a much simpler proof of the same result. We the...

Power decoding was originally developed by Schmidt, Sidorenko and Bossert for low-rate Reed–Solomon codes (RS) [1], and is usually capable of decoding almost as many errors as the Sudan decoder [2] though it is a unique decoder. When an answer is found, this is always the closest codeword, but in some cases the method will fail; in particular, this happens if two codewords are equally close to ...

space subcode of a Reed-Solomon (SSRS) code Over GF(2"') is the set of RS codewords, whose components all lie in a particular GF(2)subspace of GF(2"). SSRS codes include both generalized B C H codes and "trace-shortened" Rs codes [2][3] as special casea. In this paper we present an explicit formula for t h e dimension of an arbitrary RS subspace subcode. Using this formula, we And that in many ...

Algebraic error-correcting codes that achieve the optimal trade-off between rate and fraction of errors corrected (in the model of list decoding) were recently constructed by a careful “folding” of the Reed-Solomon code. The “low-degree” nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes useful for list decoding arise out of the Artin-F...