Classification of error locator polynomials for double error correcting BCH codes

We give a complete classification of the error locator polynomials that occur in the Berlekamp decoding of double error correcting (DEC) Bose-Chaudhuri-Hocquenghem (BCH) codes. We present a new construction showing that all quadratic error locator polynomials produced by received vectors falling in the interstitial region between decoding spheres are illegitimate and have no roots. Furthermore, we show that a small subset of received vectors in the interstitial region produce cubic error locator polynomials that are illegitimate except for the correctable case of a triple error pattern with three equally spaced errors in the cyclic sense.