Showing papers on "BCH code published in 1968"

Polynomial codes

Abstract: A class of cyclic codes is introduced by a polynomial approach that is an extension of the Mattson-Solomon method and of the Muller method. This class of codes contains several important classes of codes as subclasses, namely, BCH codes, Reed-Solomon codes, generalized primitive Reed-Muller codes, and finite geometry codes. Certain fundamental properties of this class of codes are derived. Some subclasses are shown to be majority-logic decodable.

Nonbinary BCH decoding (Abstr.)

Abstract: The decoding of BCH codes readily reduces to the solution of a certain key equation. An iterative algorithm is presented for solving this equation over any field. Following a heuristic derivation of the algorithm, a complete statement of the algorithm and proofs of its principal properties are given. The relationship of this algorithm to the classical matrix methods and the simplification which the algorithm takes in the special case of binary codes is then discussed. The generalization of the algorithm to BCH codes with a slightly different definition, the generalization of the algorithm to decode erasures as well as errors, and the extension of the algorithm to decode more than t errors in certain eases are also presented.

The Improvement of Digital HF Communication Through Coding: II--Tandem Interleaved Cyclic Coding

Abstract: In previous work the technique of error correction of digital data through the use of interleaved cyclic codes and a set of probability functions for the evaluation of error patterns have been presented. In Part I of this paper [1] the performance of a wide range of Bose-Chaudhuri-Hocquenghem (BCH) codes and PMsymbol codes was evaluated on representative portions of the data. Here a method (identified as tandem interleaved cyclic coding) will be presented which allows for a significant increase in error-rate improvement at a reduction in the delay time introduced into the channel. It is demonstrated that it is possible to get almost I00 percent error correction for delays under three seconds for all channel conditions measured.

Some lower bounds on the minimum weight of cyclic codes of composite length

Abstract: An extension of Goethals' results [3] is presented. Cyclic codes of composite block length n = n_{1}n_{2} with n_{1} and n_{2} relatively prime are considered. By using a modified form of the Mattson-Solomon formulation [5], some lower bounds on the minimum weight are obtained. These lower bounds improve on the BCH bound [4] for a considerable number of nonprimitive BCH codes.

The weight structure of some Bose-Chaudhuri codes (Corresp.)

Abstract: Absfracf-This correspondence reports on results pertaining to the min imum distance and over-all weight structure of many BoseChaudhuri-Hocquenghem (BCH) csdes generated by nonprimitive Galois field elements.‘*2*3 It will be shown that a strict interpretation of the orginal methods used to calculate a min imum distance bound for these codes leads to improvements in some previous distance estimates for particular values of code length n and number of information symbols k. Computer analysis has been used to determine the complete weight structure for many nonprimitive (BCH) codes of length up to 69. Tables are presented giving the BCH bounds and actual min imum distances for nearly all these codes. The complete weight structures are also included for many of the codes. For a number of codes the actual min imum distance exceeds the bound, in some cases by as much as 100 percent. It is observed that the inclusion of 1 as a generator polynomial root (equivalent to adding (x + 1) as a factor) can alter both the bound and actual min imum distance in various ways: no change in distance, a customary increase of one in distance, or a substantial distance increase which in some cases is double.

Performance of selected block and convolutional codes on a fading HF channel

Abstract: This paper describes the application of selected block and convolutional coding techniques to digital transmission over an HF radio channel. Performance results are presented for interleaved binary Bose-Chaudhuri-Hocquenghem (BCH) codes, interleaved two-stage concatenated codes, and diffuse convolutional codes. The performance results are based upon raw error data recorded in transmission over a 640-km HF path, with various data runs representing the typical effects of frequency-selective and non-selective fading, atmospheric impulse noise, and interference from other users of the HF band. The performance of binary BCH codes with bit interleaving is presented and the selection of a code to meet a specified performance criterion is described. The sensitivity of specific codes to changing channel conditions is presented, along with the cost, in terms of interleaving, of designing for one set of channel conditions rather than others. Concatenated codes are discussed as a means of effective error control on channels with clustered errors. Performance data are presented for selected two-stage codes used with inner-stage word interleaving and the effects of varying certain of the code parameters are shown. Emphasis is placed on code designs that use the inner-stage code only for error detection and the outer-stage code mainly for erasure filling with only a small amount of symbol error correction. The impact of error clustering on the performance of concatenated codes is discussed. Two diffuse convolutional codes are evaluated with the use of recorded error data in computer simulations of their threshold-decoding algorithms. Performance of various configurations of these convolutional codes is given and a comparison is made with that of a simple block coding scheme having the same decoder storage requirements. The various coding techniques studied are compared on the basis of achievable performance as well as certain factors affecting the feasibility of implementation.

On threshold decoding of cyclic codes

Abstract: Bose-Chaudhuri-Hocquenghem (BCH) codes are very powerful random error-correcting techniques. We have investigated whether all BCH codes can be L -step orthogonalized, and have found a specific class of double error-correcting BCH codes which cannot be L -step orthogonalized. We show further that all BCH codes with length q m − 1, where q is a power of any prime p(q = p 8 ), and all Euclidean geometry codes, can be one-step decoded by parity checks to correct a significant number of errors. These parity vectors need not be orthogonal to each other. For the general case, we have not been able to determine whether they can or cannot be decoded to their minimum distances by such a technique. The above codes decoded by nonorthogonal parity checks in the manner given herein are comparable to projective geometry codes, decoded by Rudolph's method.

On a class of wide-sense binary BCH codes whose minimum distances exceed the BCH bound (Corresp.)

Abstract: where cl is the Euclidean metric. Note that h(n) in (2) need not take the special form given in (3); it need only satisfy the requirements given in the above lemma. The limit of the estimate given in (5) indicates a relationship between a Euclidean distance and the probability of misclassification. Moreover, this limit provides a sufficient condition (upon taking the square root and multiplying by l/2) for classifying a given set of vectors from two categories within a given probability of error.

The improvement of digital hf communication through coding

Abstract: : In previous papers the technique of error correction of digital data through the use of interleaved cyclic codes and a set of probability functions for the evaluation of error patterns have been presented. In this paper the previous results are extended to a wide range of BCH and symbol codes. A set of simple equations is presented for the description of an interleaved cyclic code and its associated delay, and a method is presented which allows for a significant increase in error rate improvement at a reduction in the delay time introduced into the channel. It is demonstrated that the performance of interleaved cyclic codes is sufficient to correct all types of measured HF error patterns; that, using delay as a basis of comparison, only the total bit interleaving is important in achieving error correction; and that it is possible to get almost 100 percent error correction for delays under 3 seconds for all channel conditions measured.