On the permutation automorphisms of binary cubic codes

We investigate the structural properties of binary cubic codes. 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 $5$-dimensional binary code whose length is at least $30$ and whose permutation automorphism group is generated by a fixed point free permutation of order $3$. We also provide some computational results for the five dimensional binary cubic codes of length smaller than $30$.