Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes

We present \(O(m^3)\) algorithms for specifying the support of minimum-weight words of extended binary BCH codes of length \(n=2^m\) and designed distance \(d(m,s,i):=2^{m-1-s}-2^{m-1-i-s}\) for some values of \(m,i,s\), where \(m\) may grow to infinity. The support is specified as the sum of two sets: a set of \(2^{2i-1}-2^{i-1}\) elements, and a subspace of dimension \(m-2i-s\), specified by a basis. In some detail, for designed distance \(6\cdot 2^j\), we have a deterministic algorithm for even \(m\geq 4\), and a probabilistic algorithm with success probability \(1-O(2^{-m})\) for odd \(m>4\). For designed distance \(28\cdot 2^j\), we have a probabilistic algorithm with success probability \(\geq 1/3-O(2^{-m/2})\) for even \(m\geq 6\). Finally, for designed distance \(120\cdot 2^j\), we have a deterministic algorithm for \(m\geq 8\) divisible by \(4\). We also present a construction via Gold functions when \(2i|m\). Our construction builds on results of Kasami and Lin (IEEE T-IT, 1972), who proved that for extended binary BCH codes of designed distance \(d(m,s,i)\), the minimum distance equals the designed distance. Their proof makes use of a non-constructive result of Berlekamp (Inform. Contrl., 1970), and a constructive ``down-conversion theorem'' that converts some words in BCH codes to lower-weight words in BCH codes of lower designed distance. Our main contribution is in replacing the non-constructive argument of Berlekamp by a low-complexity algorithm. In one aspect, we extends the results of Grigorescu and Kaufman (IEEE T-IT, 2012), who presented explicit minimum-weight words for designed distance \(6\) (and hence also for designed distance \(6\cdot 2^j\), by a well-known ``up-conversion theorem''), as we cover more cases of the minimum distance. However, the minimum-weight words we construct are not affine generators for designed distance \(>6\).