On bases of BCH codes with designed distance $3$ and their extensions

We consider narrow-sense BCH codes of length $p^m-1$ over ${{\mathbb{F}}}_{p}$, $m\geqslant3$. We prove that neither such a code with designed distance $\delta=3$ nor its extension for $p\geqslant5$ is generated by the set of its codewords of the minimum nonzero weight. We establish that extended BCH codes with designed distance $\delta=3$ for $p\geqslant3$ are generated by the set of codewords of weight $5$, where basis vectors can be chosen from affine orbits of some codewords.