I came across this paper by V.I.Yorgov named "Binary self-dual codes with automorphisms of odd order". (Actually I was first reading another paper by Borello that cited this paper. It later become important that I actually understand the proof of this theorem.) Unfortunately I don't know any Russian, so I couldn't read it. I wonder if anyone could help me with this? Or anyone knows how to prove the theorem?
The paper is here:
http://www.mathnet.ru/links/64c42446ed8bceff5260b4ea78ce9c1f/ppi1198.pdf
A brief explanation of the theorem Borello is citing:
Let $\sigma$ be the automorphism of order $p$. Assume the length of the code is $n$, here $n$ is a multiple of $p$. Consider the ambient space $ V=\mathbb{F}_2^n$, then $V$ can be decomposed into $V(\sigma) \oplus V(\sigma)^\bot$, where $V(\sigma)$ is the space fixed by $\sigma$ and $V(\sigma)^\bot$ is its dual.
And $C$ can be decomposed into $C(\sigma)\oplus C(\sigma)^\bot$. Here $C(\sigma)=C \cap V(\sigma)$ and $C(\sigma)^\bot=C \cap V(\sigma)^\bot$.
On the other hand, $C$ can be divide into several orbits of $\sigma$, each one of them isomorphic to $ \mathbb{F}_2[p] \cong \mathbb{F}_2[x]/(x^p + 1) \cong \mathbb{F}_2[x]/(x+1) \oplus \mathbb{F}_2[x]/(1 + x + \cdots + x^{p-1})\cong \mathbb{F}_2 \oplus \mathbb{F}_{2^{p-1}}$. It's easy to see that $V(\sigma)$ is several copies of $ \mathbb{F}_2[x]/(x+1)$ is fixed by $\sigma$, and $V(\sigma)^\bot$ is several copies of $ \mathbb{F}_2[x]/(1 + x + \cdots + x^{p-1})$.
On $\mathbb{F}_{2^{p-1}}$, there is an involution defined by the Frobenius map $ \tau: \alpha \mapsto \alpha^{2^\frac{p-1}{2}}$. Therefore we can defined a Hermitian Conjugate on $C(\sigma)^\bot$ using this map. It turns out to be inverse the last $p-1$ bits in each orbit.
And the theorem says that if $C$ is self-dual if and only if $C(\sigma)$ is self-dual and $C(\sigma)^\bot$ is Hermitian self-dual. I assume Hermitian self-dual means its dual is its Hermitian conjugate.
I wonder if anyone knows how to prove this?
