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Showing papers on "BCH code published in 1969"


Journal ArticleDOI
TL;DR: It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback shift register capable of generating a prescribed finite sequence of digits.
Abstract: It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback shift register capable of generating a prescribed finite sequence of digits. The shift-register approach leads to a simple proof of the validity of the algorithm as well as providing additional insight into its properties. The equivalence of the decoding problem for BCH codes to a shift-register synthesis problem is demonstrated, and other applications for the algorithm are suggested.

2,269 citations


Journal ArticleDOI
TL;DR: This paper is a compendium of results based on a simple observation: two information symbols can be appended to certain nonbinary BCH codes without affecting the guaranteed minimum distance of these codes.
Abstract: This paper is a compendium of results based on a simple observation: two information symbols can be appended to certain nonbinary BCH codes without affecting the guaranteed minimum distance of these codes. We give two formulations which achieve this result; the second yields information regarding the weights of coset leaders for the original BCH codes. Single-error-correcting Reed-Solomon codes with the added information symbols yield perfect codes for the Hamming metric. We use these lengthened Reed-Solomon codes as building blocks for perfect single-error-correcting codes in another metric.

68 citations


Journal ArticleDOI
TL;DR: It is shown that if m eq 8, 12 and m > 6 , there are some binary primitive BCH codes of length 2^{m} - 1 whose minimum weight is greater than the BCH bound.
Abstract: It is shown that if m eq 8, 12 and m > 6 , there are some binary primitive BCH codes (BCH codes in a narrow sense) of length 2^{m} - 1 whose minimum weight is greater than the BCH bound. This gives a negative answer to the question posed by Peterson [1] of whether or not the BCH bound is always the actual minimum weight of a binary primitive BCH code. It is also shown that for any even m \geq 6 , there are some binary cyclic codes of length 2^{m} - 1 that have more information digits than the primitive BCH codes of length 2^{m} - 1 with the same minimum weight.

64 citations



Journal ArticleDOI
TL;DR: An upper bound on the transmission ratio k/n for binary cyclic codes whose extended codes are invariant under the affine group of permutations, is presented and is shown to approach zero as the code length n increases.
Abstract: An upper bound on the transmission ratio k/n for binary cyclic codes whose extended codes are invariant under the affine group of permutations, is presented. As a consequence, the transmission ratio k/n of any affine-invariant code with a fixed d (minimum weight)/ n is shown to approach zero as the code length n increases. This is an extension of the Lin and Weldon result for primitive BCH codes.

20 citations


Journal ArticleDOI
TL;DR: This work determines those codes (as specified by n, the length of the code, and not by k ) whose minimum-distance lower bound doubles when the root one is added to the code generator.
Abstract: We consider the problem of determining which irreducible polynomials with coefficients in a finite field GF(q) are quasi-self-reciprocal. Our characterization of these polynomials is in terms of a set of positive integers, Dq . Dq has applications to certain BCH codes. This work determines those codes (as specified by n , the length of the code, and not by k ) whose minimum-distance lower bound doubles when the root one is added to the code generator.

14 citations


Journal ArticleDOI
W. Gore1
TL;DR: A necessary condition for the existence of a set of J orthogonal parity check equations is developed, and it is demonstrated that, except for the trivial codes, the Reed-Solomon codes are not L -step Orthogonalizable.
Abstract: A necessary condition for the existence of a set of J orthogonal parity check equations is developed, and it is demonstrated that, except for the trivial codes, the Reed-Solomon codes are not L -step orthogonalizable. Massey's concept of threshold decoding is generalized, and it is demonstrated that the Reed-Solomon codes are completely threshold decodable. Since every BCH code is a subcode of some Reed-Solomon code, every BCH code has a generalized threshold decoder.

8 citations


Journal ArticleDOI
TL;DR: The class of polynomial codes introduced by Kasami, Lin, and Peterson has considerable inherent algebraic and geometric structure as mentioned in this paper, and it contains many well known classes of codes as subclasses, such as BCH codes and geometry codes.
Abstract: The class of polynomial codes introduced by Kasami, Lin, and Peterson has considerable inherent algebraic and geometric structure. It contains many well known classes of codes as subclasses, such as BCH codes and geometry codes. The purpose of this paper is to derive further properties of polynomial codes. It is hoped that these properties may impart more algebraic structure to BCH codes and geometry codes. Firstly, combinatorial expressions for enumerating the number of information digits of certain subclasses of polynomial codes are derived. Secondly, the exact minimum distance of a subclass of BCH codes is established and a tight BCH bound on the minimum distance of the dual of a polynomial code is obtained. Finally, it is shown that a primitive polynomial code is a subcode of the m th power of an extended primitive BCH code where the m th power of a code is defined as the direct-product of a code with itself m times.

7 citations


01 Oct 1969
TL;DR: It is shown that, for all practical purposes, one can synthesize codes of given length and minimum distance by a simple root-distance relation, which gives the strong relation between the arithmetic code and the BCH code.
Abstract: : It is shown that, for all practical purposes, one can synthesize codes of given length and minimum distance by a simple root-distance relation. This gives the strong relation between the arithmetic code and the BCH code.

4 citations


Journal ArticleDOI
TL;DR: It is shown that the proposed decoding scheme can be applied to several BCH codes making it possible to correct many errors beyond the ones guaranteed by the known minimum distance and also the codes will be "effectively" majority decodable.
Abstract: A decoding scheme is given for some block codes for which it is known how to decode a subcode. It is shown that the proposed decoding scheme can be applied to several BCH codes making it possible to correct many errors beyond the ones guaranteed by the known minimum distance and also the codes will be "effectively" majority decodable.

2 citations