A View of Gaussian Elimination Applied to Early Stopped Berklekamp-Massey Algorithm

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we adopt a restricted Gaussian elimination on the Hankel structured augmented syndrome matrix to reinterpret an early stopped version of the Berlekamp–Massey algorithm in which only <formula formulatype="inline"> <tex>$(t + e)$</tex></formula> iterations need to be performed for the decoding of BCH codes up to <formula formulatype="inline"><tex>$t$</tex></formula> errors, where <formula formulatype="inline"><tex>$e$</tex></formula> is the number of errors actually occurring with <formula formulatype="inline"><tex>$e \leq t$</tex></formula> , instead of the <formula formulatype="inline"><tex>$2t$</tex> </formula> iterations required in the conventional Berklekamp–Massey algorithm. The minimality of <formula formulatype="inline"> <tex>$(t + e)$</tex></formula> iterations in this early stopped Berklekamp–Massey (ESBM) algorithm is justified and related to the subject of simultaneous error correction and detection in this paper. We show that the multiplicative complexity of the ESBM algorithm is upper bounded by <formula formulatype="inline"><tex>$(te+ e^{2} - 1) \forall e \leq t$</tex></formula> and except for a trivial case, the ESBM algorithm is the most efficient algorithm for finding the error-locator polynomial. </para>