We investigate the structural properties of binary linear codes whose permutation automorphism group has a fixed point free automorphism of order $3$. We prove that up to dimension or codimension $4$, there is no binary linear code whose permutation automorphism group is generated by a fixed point free permutation of order $3$. We also prove that there is no binary $5$-dimensional linear code whose length is at least $30$ and whose permutation automorphism group is generated by a fixed point free permutation of order $3$.

翻译：我们研究了置换自同构群包含一个三阶无不动点自同构的二元线性码的结构性质。我们证明，在维数或余维数不超过4的情况下，不存在置换自同构群由三阶无不动点置换生成的二元线性码。我们还证明，不存在长度至少为30且置换自同构群由三阶无不动点置换生成的五维二元线性码。