Showing papers on "BCH code published in 1979"
IBM1
TL;DR: By using the theory of finite field Fourier transforms, the subject of error control codes is described in a language familiar to the field of signal processing and several alternative encoder/ decoder schemes are described by frequency domain reasoning.
Abstract: By using the theory of finite field Fourier transforms, the subject of error control codes is described in a language familiar to the field of signal processing The many important uses of spectral techniques in error control are summarized Many classes of linear codes are given a spectral interpretation and some new codes are describe Several alternative encoder/ decoder schemes are described by frequency domain reasoning In particular, an errors-and-erasures decoder for a BCH code is exhibited which has virtually no additional computations over an errors-only decoder Techniques for decoding BCH, RS, and alternant codes (Goppa codes) a short distance beyond the designed distance are discussed Also, a modification to the definition of a BCH code is described which reduces the decoder complexity without changing the code's rate or minimum distance
219 citations
TL;DR: In this paper, the construction of BCH codes over finite fields was shown to be similar to that for BCH code over finite integer rings, where the generator polynomials are derived by factorization of x n −1 over the unit ring of an appropriate extension of the Finite integer ring.
Abstract: Bose-Chadhuri-Hocquenghem (BCH) codes with symbols from an arbitrary finite integer ring are derived in terms of their generator polynomials. Tile derivation is based on the factorization of x^{n}-1 over the unit ring of an appropriate extension of the Finite integer ring. The construction is thus shown to be similar to that for BCH codes over finite fields.
122 citations
TL;DR: The algorithm for developing the rational approximations is based on continued fraction techniques and is virtually equivalent to an algorithm employed by Berlekamp for decoding BCH codes.
Abstract: Theorems are presented concerning the optimality of rational approximations using non-Archimedean norms. The algorithm for developing the rational approximations is based on continued fraction techniques and is virtually equivalent to an algorithm employed by Berlekamp for decoding BCH codes. Several variations of the continued fraction technique and Berlekamp's algorithm are illustrated on a common example.
102 citations
Patent•
31 Dec 1979
TL;DR: In this article, a method and apparatus for detecting errors in a data word and for correcting up to two error bits therein through using Bose-Chaudhuri-Hocquenghem (BCH) codes is presented.
Abstract: A method and apparatus for detecting errors in a data word and for correcting up to two error bits therein through using Bose-Chaudhuri-Hocquenghem (BCH) codes. A Galois field table is arranged in an anti-log and log format to facilitate the calculation of certain values such as ##EQU1## associated with the BCH codes.
25 citations
TL;DR: A new scheme for reducing the numerical complexity of the standard B.C.H. and Reed–Solomon (R.S.) decoding algorithms is developed and the process of calculating syndromes over GF(2m) is shown to require only a small fraction of the number of multiplications and additions that is required by using standard methods.
Abstract: A new scheme for reducing the numerical complexity of the standard B.C.H. and Reed–Solomon (R.S.) decoding algorithms is developed. Specifically, the process of calculating syndromes over GF(2m) is shown to require only a small fraction of the number of multiplications and additions that is required by using standard methods. As an example, the calculation of the 32 syndromes of the (255, 223, 33) Reed–Solomon code (NASA standard for concatenation with convolutional codes) is shown to require 90% fewer multiplications and 78% fewer additions than the conventional method of computation. A computer simulation also verifies these results.
7 citations
TL;DR: Using continued fractions, a simplified algorithm for decoding B.C.H. and R.S. codes is developed that corrects both erasures and errors on a finite field GF(qm) that is both simpler to understand and to implement than more conventional algorithms.
Abstract: Using continued fractions, a simplified algorithm for decoding B.C.H. and R.S. codes is developed that corrects both erasures and errors on a finite field GF(qm). The decoding method is a modification of the Forney-Belekamp technique. It is believed that the present scheme is both simpler to understand and to implement than more conventional algorithms.
7 citations
TL;DR: The main purpose of this paper is to report on the implementation of the error locator polynomial, and the Chien's search for finding the error locations, using the table look-up technique suitable to a microprocessor based decoder.
Abstract: Presents some preliminary results on the implementation of a binary BCH decoder involving a combination of specially designed hardware for some operations and a software implementation on a microprocessor for the remaining operations. With such a microprocessor based decoder, a large number of different BCH codes can be accommodated with little or no hardware changes. The decoding task is a function of the software and thus allows common hardware to support a wide range of functions. The main purpose of this paper is to report on the implementation of the error locator polynomial, and the Chien's search for finding the error locations, using the table look-up technique suitable to a microprocessor based decoder.
5 citations
TL;DR: A new class of error-correcting codes, which generalizes the BCH codes and the polynomial codes of Goethals is constructed, which gives a new construction of orthogonal idempotents, and hence of minimal ideals, in any semisimple commutative algebra.
Abstract: A new class of error-correcting codes, which generalizes the BCH codes and the polynomial codes of Goethals is constructed. These new codes have associated designed distances, which give lower bounds for their error-correcting capability. We also give a new construction of orthogonal idempotents, and hence of minimal ideals, in any semisimple commutative algebra.
4 citations
TL;DR: It is proved that the covering radius of a code is defined as the smallest integer such that the union of the spheres of radius about the codewords equals the containing space.
Abstract: Gorenstein, Peterson, and Zierler have conjectured that no t -error-correcting BCH code of length 2^{m} - 1 with t > 2 is quasi-perfect. This conjecture is proved. The covering radius of a code is defined as the smallest integer \rho such that the union of the spheres of radius ia about the codewords equals the containing space.
3 citations
TL;DR: In this paper, a category Bch(X) of behaviours is introduced as a formalization of an external description of systems, and it is shown that the forgetful functor W: Bch (x) → K2 is algebraic and therefore Bch has, and W preserves, whatever limits K has.
Abstract: A category Bch(X) of behaviours is introduced as a formalization of an external description of systems. It is show that the forgetful functor W: Bch(x) → K2 is algebraic and that therefore Bch(x) has, and W preserves, whatever limits K has. Colimits and an image factorization system for Bch(X) are also investigated.