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

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 $(t + e)$ iterations need to be performed for the decoding of BCH codes up to $t$ errors, where $e$ is the number of errors actually occurring with $e \leq t$ , instead of the $2t$ iterations required in the conventional Berklekamp–Massey algorithm. The minimality of $(t + e)$ 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 $(te+ e^{2} - 1) \forall e \leq t$ and except for a trivial case, the ESBM algorithm is the most efficient algorithm for finding the error-locator polynomial.