A binary linear code whose permutation automorphism group has a fixed point free permutation of order $3$ is called a binary cubic code. The scope of this paper is to investigate the structural properties of binary cubic codes. Let $C$ be a binary cubic $[n,k]$ code. In this paper, we prove that if $n\geq 30$ and $C$ has permutation automorphism group of order three, then $k\geq 6$. Additionally, we show that if $n < 30$ and $k\leq 4$, then the permutation automorphism group of $C$ has order greater than three. Moreover, along the way, we provide some results on the structure of the higher dimensional cubic codes. In particular, we present some results concerning the structure of the putative extremal self-dual $[72,36,16]$ code under the assumption that it is cubic.

翻译：若一个二元线性码的置换自同构群包含一个无不动点的3阶置换，则称该码为二元三次码。本文旨在研究二元三次码的结构性质。设$C$为一个二元三次$[n,k]$码。本文证明，若$n\geq 30$且$C$的置换自同构群阶数为三，则$k\geq 6$。此外，我们证明若$n < 30$且$k\leq 4$，则$C$的置换自同构群阶数大于三。同时，在研究过程中，我们给出了关于高维三次码结构的一些结果。特别地，在假设极值自对偶$[72,36,16]$码为三次码的前提下，我们提出了关于其结构的若干结果。