We consider narrow-sense BCH codes of length p^m − 1 over \({{\mathbb{F}}}_{p}\), m ≥ 3. We prove that neither such a code with designed distance δ = 3 nor its extension for p ≥ 5 is generated by the set of its codewords of the minimum nonzero weight. We establish that extended BCH codes with designed distance δ = 3 for p ≥ 3 are generated by the set of codewords of weight 5, where basis vectors can be chosen from affine orbits of some codewords.

中文翻译：

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基于设计距离为3的BCH码及其扩展

我们考虑的长度狭义BCH码p^米 - 1之上\（{{\ mathbb {F}}} _ {P} \） ，米≥3.我们证明，无论这样的设计距离的码δ  = 3也不其扩展p ≥5由该组中的最小非零重量的其码字的生成。我们建立扩展BCH码设计距离δ  = 3 p ≥3通过设定的权重5，其中的基矢量可以从一些码字的仿射轨道被选择的码字的生成。