Error Locator Polynomial Research Articles

Parallel implementation of the Schur Berlekamp-Massey algorithm on a linearly connected processor array

The Berlekamp-Massey algorithm (BMA) (E. Berlekamp, 1968; J. Massey, 1969) is important in the decoding of Reed-Solomon (RS), and more generally, Bose-Chaudhuri-Hocquenghem (BCH) block error control codes. For a t-error correcting code the BMA has time complexity O(t/sup 2/) when implemented on a sequential computer. However, the BMA does not run efficiently on a parallel computer. The BMA can be mapped into the Schur BMA. The paper presents the implementation of the BMA and Schur BMA together on a linearly connected array of 2t processors. The resulting machine computes the error locator polynomial with a time complexity of O(t). >

Simple algorithms for BCH decoding

Proposes some simple algorithms for decoding BCH codes. The authors show that the pruned FFT is an effective method for evaluating syndromes and for finding the roots of error-locator polynomials. They show that a simple variation of the basic Gaussian elimination procedure can be adapted to compute the error-locator polynomial efficiently for codes with small designed distance. Finally, they give a procedure for computing the error values that has half the complexity of the Forney algorithm. >

Decoding double-error-correcting Reed-Solomon codes

The decoding of double-error-correcting (DEC) Reed-Solomon (RS) codes is considered. It is shown that by modifying a well known decoding algorithm for DEC RS codes and solving the error-locator polynomial by Berlekamp's method for solving quadratic equations, efficient hardware architectures can be derived. Furthermore, these architectures are particularly suited to implementation over the dual basis. As an example, the architecture of a (15, 11) RS codec is described. The approaches discussed here also lend themselves to the decoding of double-error-correcting/triple-error-detecting RS codes and allow for reduced decoding times compared with alternative approaches to decoding these codes.

New time domain errors and erasures decoding algorithm for BCH codes

A new algorithm is presented for solving the key equation that simultaneously computes the error locator polynomial and the errata evaluator polynomial. The algorithm is similar to the Berlekamp algorithm but is more symmetrical in its treatment of the iterated pairs of polynomials making it particularly well suited to a highly parallel hardware implementation.

Fast decoding of codes from algebraic curves

The author shows how the fast decoding algorithm of Justesen et al. (1989), for codes from algebraic plane curves, can be extended such that codes from curves in an r-dimensional space can be decoded. He shows how Sakata's (1990) Berlekamp-Massey (1969) extension can be used to find an error locator polynomial, and also shows that the cost of doing this increases with the dimension of the space. Unfortunately, the error-correcting capability gets worse for codes from curves in higher dimensional spaces. For a specific curve, he determines the error values faster than by solving linear equations. The method is an extension of the transformation method known from cyclic codes. >

A new procedure for decoding cyclic and BCH codes up to actual minimum distance

The paper presents a new procedure for decoding cyclic and BCH codes up to their actual minimum distance. It generalizes the Peterson decoding procedure and the procedure of Feng and Tzeng (1991) using nonrecurrent syndrome dependence relations. For a code with actual minimum distance d to correct up to t=[(d-1)/2] errors, the procedure requires a (2t+1)/spl times/(2t+1) syndrome matrix with known syndromes above the minor diagonal and unknown syndromes and their conjugates on the minor diagonal. In contrast to previous procedures, this procedure is primarily aimed at solving for the unknown syndromes instead of determining an error-locator polynomial. Decoding is then accomplished by determining the error vector as the inverse Fourier transform of the syndrome vector (S/sub 0/, S/sub 1/, S/sub n-1/). The authors show that with this procedure, all binary cyclic and BCH codes of length >

Use of Grobner bases to decode binary cyclic codes up to the true minimum distance

A general algebraic method for decoding all types of binary cyclic codes is presented. It is shown that such a method can correct t=[(d-1)/2] errors, where d is the true minimum distance of the given cyclic code. The key idea behind this decoding technique is a systematic application of the algorithmic procedures of Grobner bases to obtain the error-locator polynomial L(z). The discussion begins from a set of syndrome polynomials F and the ideal T(F) generated by F. It is proved here that the process of transforming F to the normalized reduced Grobner basis of I(F) with respect to the purely lexicographical ordering automatically converges to L(z). Furthermore, it is shown that L(z) can be derived from any normalized Grobner basis of I(F) with respect to any admissible total ordering. To illustrate this new approach, the procedures for decoding certain BCH codes and quadratic residue codes are demonstrated. >

The n-dimensional key equation and a decoding application

The author introduce the n-dimensional key equation, which exhibits the error-locator polynomial of an n-dimensional cyclic code as a product of n univariate polynomials and the error-evaluator polynomial as an n-variable polynomial. They then reinterpret these polynomials in the context of linear recurring sequences. In particular, they reduce the decoding problem to successive application of the Berlekamp-Massey algorithm. With this new method, they are able to decode (up to half their minimum distance) many codes in a table of 2-D cyclic codes due to Jensen (1985).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On Justesen’s algebraic geometry codes

An isomorphism preserving Hamming distance between two algebraic geometry (AG) codes is presented to obtain the main parameters of Justesen’s algebraic geometry (JAG) codes. To deduce a simple approach to the decoding algorithm, a code word in a “small” JAG code is used to correspond to error-locator polynomial. By this means, a simple decoding procedure and its ability of error correcting are explored obviously. The lower and upper bounds of the dimension of AG codes are also obtained.

The algebraic decoding of the (41, 21, 9) quadratic residue code

A new algebraic approach for decoding the quadratic residue (QR) codes, in particular the (41, 21, 9) QR code, is presented. The key ideas behind this decoding technique are a systematic application of the Sylvester resultant method to the Newton identities associated with the syndromes to find the error-locator polynomial, and next a method for determining error locations by solving certain quadratic, cubic, and quartic equations over GF(2/sup m/) in a new way which uses Zech's logarithms for the arithmetic. The logarithms developed for Zech's logarithms save a substantial amount of computer memory by storing only a table of Zech's logarithms. These algorithms are suitable for implementation in a programmable microprocessor or special-purpose VLSI chip. It is expected that the algebraic methods developed can apply generally to other codes such as the BCH and Reed-Solomon codes. >

Closed solution of Berlekamp's algorithm for fast decoding of BCH codes

We present a decision tree solution for the most complicated step in decoding binary BCH codes, namely the computation of an error location polynomial over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) from the syndrome vector of received data. We run Lin's iterative version of the Berlekamp-Massey algorithm symbolically, keeping the results at each level in the form of branches of a binary decision tree. A decoder can then be constructed that uses the derived formulas to evaluate a decision variable at each level. Complete traversal of the tree using the decision variables leads to the correct polynomial coefficients for the received vector. The decoder can be implemented in a very straightforward way with a simple processor or program that performs extension field arithmetic, or it can be realized entirely in hardware using lookup tables for multiplications, inverses, and exponents, and exclusive OR operations for addition. This latter method can provide an extremely fast decoder that is compatible with realization in VLSI.

Finding BCH error locator polynomials in one step

Syndrome polynomials produced in a BCH decoder span an ideal in a multivariate polynomial ring. Because the error locators satisfy every member of the ideal, polynomial ideal theory is used to find another, more easily solvable system of equations having the same roots. One such equation is the (univariate) error locator polynomial. The Letter shows how to write this polynomial in one mathematical step from the syndrome polynomials.

A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes

A generalization of the Berlekamp-Massey algorithm is presented for synthesizing minimum length linear feedback shift registers for generating prescribed multiple sequences. A more general problem is first considered, that of finding the smallest initial set of linearly dependent columns in a matrix over an arbitrary field, which includes the multisequence problem as a special case. A simple iterative algorithm, the fundamental iterative algorithm (FIA), is presented for solving this problem. The generalized algorithm is then derived through a refinement of the FIA. Application of this generalized algorithm to decoding cyclic codes up to the Hartmann-Tzeng (HT) bound and Roos bound making use of multiple syndrome sequences is considered. Conditions for guaranteeing that the connection polynomial of the shortest linear feedback shift register obtained by the algorithm will be the error-locator polynomial are determined with respect to decoding up to the HT bound and special cases of the Roos bound. >

Algebraic decoding of the (32, 16, 8) quadratic residue code

An algebraic decoding algorithm for the 1/2-rate (32, 16, 8) quadratic residue (QR) code is found. The key idea of this algorithm is to find the error locator polynomial by a systematic use of the Newton identities associated with the code syndromes. The techniques developed extend the algebraic decoding algorithm found recently for the (32, 16, 8) QR code. It is expected that the algebraic approach developed here and by M. Elia (1987) applies also to longer QR codes and other BCH-type codes that are not fully decoded by the standard BCH decoding algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Decoding the triple-error-correcting (15, 5) binary BCH code by the analytic solution of the cubic error-locator polynomial over GF(24)

This paper describes a decoding algorithm for the triple-error-correcting (15, 5) binary BCH code based on the analytic solution of the cubic error-locator polynomial over the extension field GF(24). It is shown that the error-location numbers can be determined directly from the coefficients of the error-locator polynomial. The algorithm avoids the need for a Chien-type search and requires significantly fewer mathematical operations over GF(24) than a Chien-type search.

Algebraic decoding beyond e/sub BCH/ of some binary cyclic codes, when e&amp;lt;e/sub BCH/

For a number of binary cyclic codes with e>e/sub BCH/, algebraic algorithms are given to find the error locator polynomial. Thus, for these codes more errors can be corrected algebraically than by the Berlekamp-Massey algorithm. In some cases, all error patterns of weight up to e can be decoded; in other cases, only error patterns of weight up to e' with e/sub BCH/<e'<or=e can be decoded. The correctness of three of these algorithms is (partly) based on an exhaustive computer search; in all other cases, the algebraic proof is given in detail. It seems likely that many more cyclic codes can be decoded with these methods.<<ETX>>

Decoding of redundant residue polynomial codes using Euclid's algorithm

A decoding method is proposed for the redundant residue polynomial codes, a class that includes Reed-Solomon codes. The method is based on properties of Euclid's algorithm. The advantage of the method is that the computation of both the error-locator polynomial and the error-evaluator polynomial needed in conventional decoding methods can be avoided. The method is suitable for decoding concatenated codes whose outer codes are redundant residue polynomial codes, since they are easily decoded by ignoring erasures detected in the inner codes. >

Simplified procedure for correcting both errors and erasures of Reed-Solomon code using Euclidean algorithm

It is well known that the Euclidean algorithm or its equivalent, continued fractions, can be used to find the error locator polynomial and the error evaluator polynomial in Berlekamp&apos;s key equation that is needed to decode a Reed-Solomon (RS) code. In the paper, a simplified procedure is developed and proved to correct erasures as well as errors by replacing the initial condition of the Euclidean algorithm by the erasure locator polynomial and the Forney syndrome polynomial. By this means, the errata locator polynomial and the errata evaluator polynomial can be obtained simultaneously and simply, by the Euclidean algorithm only. With this improved technique, the complexity of time-domain Reed-Solomon decoders for correcting both errors and erasures is reduced substantially from previous approaches. As a consequence, decoders for correcting both errors and erasures of RS codes can be made more modular, regular, simple, and naturally suitable for both VLSI and software implementation. An example illustrating this modified decoding procedure is given for a (15, 9) RS code.

Decoding of DBEC-TBED Reed-Solomon Codes

A problem in designing semiconductor memories is to provide some measure of error control without requiring excessive coding overhead or decoding time. In LSI and VLSI technology, memories are often organized on a multiple bit (or byte) per chip basis. For example, some 256K bit DRAM's are organized in 32K × 8 bit-bytes. Byte-oriented codes such as Reed-Solomon (RS) codes can provide efficient low overhead error control for such memories. However, the standard iterative algorithm for decoding RS codes is too slow for these applications. In this correspondence we present a special decoding technique for double-byte-error-correcting (DBEC), triple-byte-error-detecting (TBED) RS codes which is capable of high-speed operation. This technique is designed to find the error locations and the error values directly from the syndrome without having to use the iterative algorithm to find the error locator polynomial.

A VLSI Design of a Pipeline Reed-Solomon Decoder

A pipeline structure of a transform decoder similar to a systolic array is developed to decode Reed-Solomon (RS) codes. An important ingredient of this design is a modified Euclidean algorithm for computing the error-locator polynomial. The computation of inverse field elements is completely avoided in this modification of Euclid's algorithm. The new coder is regular and simple, and naturally suitable for VLSI implementation. An example illustrating both the pipeline and systolic array aspects of this decoder structure is given for a RS code.