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Algebraic properties of BCH codes useful for decoding

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<mark>Journal publication date</mark>1989
<mark>Journal</mark>International Journal of Satellite Communications
Issue number3
Volume7
Number of pages5
Pages (from-to)225-229
Publication StatusPublished
<mark>Original language</mark>English

Abstract

In this paper theorems are presented which allow the simplified decoding of (n, k, δ) BCH codes in certain cases of practical interest. Such results are in a way implicit in the theory of BCH codes, but so far have not appeared explicitly in the literature. It is shown that any t0 errors, 1 ⩽ t0 ⩽ δ-1, can be detected by using any set of only t0 consecutive coefficients of the syndrome polynomial. The correction of any t0 errors, 1 ⩽ t0 ⩽ [(δ-1)/2], can be performed by using any set of 2t0 consecutive coefficients of the syndrome polynomial, where [x] means the integer part of x. Similar results are derived for punctured BCH codes. In this case sets of t0 or 2t0 consecutive coefficients, respectively, for detecting or correcting t0 errors, are selected from the δ-1-p higher-order coefficients of the modified syndrome polynomial, where p is the number of digits punctured from a code word. These results hold true even when the punctured digits are not consecutive.