Quadratic Residue Code

The algebraic decoding of binary quadratic residue codes can be performed using the Peterson or the Berlekamp-Massey algorithm once certain unknown syndromes are determined or eliminated. The technique of determining unknown syndromes is applied to the nonbinary case to decode the expurgated ternary quadratic residue code of length 23

An efficient combination between Berlekamp-Massey and Hartmann Rudolph algorithms to decode BCH codes

In digital communication and storage systems, the exchange of data is achieved using a communication channel which is not completely reliable. Therefore, detection and correction of possible errors are required by adding redundant bits to information data. Several algebraic and heuristic decoders were designed to detect and correct errors. The Hartmann Rudolph (HR) algorithm enables to decode a sequence symbol by symbol. The HR algorithm has a high complexity, that's why we suggest using it partially with the algebraic hard decision decoder Berlekamp-Massey (BM). In this work, we propose a concatenation of Partial Hartmann Rudolph (PHR) algorithm and Berlekamp-Massey decoder to decode BCH (Bose-Chaudhuri-Hocquenghem) codes. Very satisfying results are obtained. For example, we have used only 0.54% of the dual space size for the BCH code (63,39,9) while maintaining very good decoding quality. To judge our results, we compare them with other decoders

Developing Efficient Algorithms of Decoding the Systematic Quadratic Residue Code with Lookup Tables

The lookup table methods for decoding binary systematic Quadratic Residue (QR) code are presented in this paper. The key ideas behind this decoding technique are based on one to one corresponding mapping between the syndromes and the correctable error patterns. Such algorithms determine the error locations directly by lookup tables without the operations of addition and multiplication over a finite field. Moreover, the methods to dramatically reduce the memory requirement by shift-search decoding are utilized. Two new algorithm have been verified through a software simulation in C language. The new approach is modular, regular and naturally suitable for System on Chip (SOC) software implementation

On Decoding of Quadratic Residue Codes

A binary Quadratic Residue(QR) code of length n is an (n, (n+1)/2) cyclic code over GF(2m) with generator polynomial g(x) where m is some integer. The length of this code is a prime number of the form n = 8l + 1 where l is some integer. The generator polynomial g(x) is defined by g(x)=∏_(i∈Q_n) (x-βi ) where β is a primitive nth root of unity in the finite field GF(2m) with m being the smallest positive integer such that n|2m-1 and Qn is the collection of all nonzero quadratic residues modulo n given by Qn={i│i≡j2 mod n for 1≤j≤n-1}. Algebraic approaches to the decoding of the quadratic residue (QR) codes were studied in [2], [3], [4], [5], [6] and [13]. Here, in this thesis, some new more general properties are found for the syndromes of the subclass of binary QR codes of length n = 8m + 1 or n = 8m - 1. A new algebraic decoding algorithm for the (41, 21, 9) binary QR code is presented by having the unknown syndrome S3 which is a necessary condition for decoding the (41, 21, 9) QR code

On the Shape of the General Error Locator Polynomial for Cyclic Codes

General error locator polynomials were introduced in 2005 as an alternative decoding for cyclic codes. We now present a conjecture on their sparsity, which would imply polynomial-time decoding for all cyclic codes. A general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several results for specific code families, providing evidence to our conjecture. From these, a theoretical justification of the sparsity of general error locator polynomials is obtained for all binary cyclic codes with t <= 2 and n < 105, as well as for t = 3 and n < 63, except for some cases where the conjectured sparsity is proved by a computer check. Moreover, we summarize all related results, previously published, and we show how they provide further evidence to our conjecture. Finally, we discuss the link between our conjecture and the complexity of bounded-distance decoding of the cyclic codes