An error-trapping decoder for nonbinary cyclic codes (Corresp.)

Abstract

The error-trapping decoder is the simplest way of decoding cyclic codes satisfying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R &lt; 1 / t</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> is the maximum number of errors to be corrected and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> is the code rate. These codes have low rates and/or correct only a few errors. Kasami has used the concept of covering polynomials to demonstrate modified error-trapping decoders for several binary cyclic codes not satisfying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R&lt;1/t</tex> . In this paper Kasami's decoder is modified further for correcting multiple symbol errors on nonbinary cyclic codes satisfying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R &lt; 2 / t</tex> . The Berlekamp decoder for these codes requires Galois field multiplication and division of two variables which are difficult to implement. Our decoder does not require these multiplications and divisions. Further, for all double-error-correcting codes, and triple-error-correcting codes with rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R &lt; 2/3</tex> , an algorithm is presented for finding a minimum set of covering monomials.