  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 $>6$.
