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?



Need help with paper written in Russian... Yorgov's paper on self-dual codes with automorphisms of odd order
[According to MathSciNet](http://www.ams.org/mathscinet-getitem?mr=754686), a translated version of the paper is available in the journal *Problems of Information Transmission* ISSN: 0032-9460. Your local librarian should be able to help you track down a paper copy of that journal (as far as Google can tell me, there are no digital copies from prior to Springer's acquisition of it in the 2000s). 


