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 $>6$.

翻译：我们提出了$O(m^3)$算法，用于描述长度为$n=2^m$、设计距离为$d(m,s,i):=2^{m-1-s}-2^{m-1-i-s}$的扩展二进制BCH码中最小重量词的支持域，其中$m,i,s$取特定值且$m$可趋向无穷。该支持域被表示为两个集合之和：一个包含$2^{2i-1}-2^{i-1}$个元素的集合，以及一个由基指定的维度为$m-2i-s$的子空间。具体而言，对于设计距离$6\cdot 2^j$，当$m\geq 4$为偶数时我们给出确定性算法，当$m>4$为奇数时给出成功概率为$1-O(2^{-m})$的概率算法。对于设计距离$28\cdot 2^j$，当$m\geq 6$为偶数时我们给出成功概率$\geq 1/3-O(2^{-m/2})$的概率算法。最后，对于设计距离$120\cdot 2^j$，当$m\geq 8$且能被4整除时我们给出确定性算法。我们还通过Gold函数给出了$2i|m$情形下的构造方法。该构造基于Kasami和Lin（IEEE T-IT, 1972）的结果，他们证明了对于设计距离为$d(m,s,i)$的扩展二进制BCH码，最小距离等于设计距离。其证明利用了Berlekamp（Inform. Contrl., 1970）的非构造性结果，以及一个构造性的“向下转换定理”，该定理可将BCH码中的某些码字转换为更低设计距离BCH码中具有更低重量的码字。我们的主要贡献在于用低复杂度算法替代了Berlekamp的非构造性论证。在某个方面，我们的结果扩展了Grigorescu和Kaufman（IEEE T-IT, 2012）的工作，后者给出了设计距离$6$的显式最小重量码字（因此通过著名的“向上转换定理”，也适用于设计距离$6\cdot 2^j$），因为我们覆盖了更多最小距离情形。然而，对于设计距离$>6$的情况，我们构造的最小重量码字并非仿射生成元。