4
$\begingroup$

In coding theory, we know that if you take the function

\begin{equation} \alpha_q(\delta) := \limsup_{n \rightarrow \infty} \ \max \{ R(C) \mid \delta(C) \ge \delta \mid C \subseteq \mathbb{F}_q^n\}. \end{equation}

then the function is continuous. Here $R(C)$ is the rate of a code $C$ and $\delta(C)$ the relative minimum distance. This follows by spoiling a code appropriately to have codes in the vicinity of any point in the graph. A proof is in Tsfasman and Vladut's "Algebraic-geometric codes" book, for example.

I think the original proof is by Manin, but I am not sure (would like to know!)

Now if I restrict only to cyclic codes and define a similar function as the following:

\begin{equation} \alpha_q^{\text{cyclic}}(\delta) := \limsup_{n \rightarrow \infty} \ \max \{ R(C) \mid \delta(C) \ge \delta \mid C \text{ is cyclic} \}. \label{eq:defi_of_Alp_cyc} \end{equation}

Is it known that this function is also continuous? The usual proof for the general case cannot be extended (as far as I could tell).

EDIT: This question is an open problem, because it proves an open problem.

However, I am now changing my demand. Instead of 'cyclic', I can choose other class of codes like 'Goppa' codes or 'transitive' codes and make similar constructions $\alpha_q^{\text{Goppa}}$ or $\alpha_q^{\text{trans}}$ for instance. Are any such functions known to be continuous?

$\endgroup$
  • $\begingroup$ On a second thought, I think proving it to be continuous will solve the long-standing "good cyclic codes" conjecture. $\endgroup$ – Breakfastisready Jan 10 at 21:54
4
$\begingroup$

To answer the question on the origin of $\alpha_q$. The original proof goes back to Manin's article "What is the maximum number of points on a curve over $\mathbb F_2$", J. Fac. Sci. Univ. Tokyo, 1981. Manin and Marcolli have some more recent articles ([1], [2]) studying $\alpha_q$ from a sort of a statistical mechanical perspective.

$\endgroup$
  • $\begingroup$ I have changed my question now so I cannot accept this answer, but thank you for telling me about this paper (and also providing the full text) $\endgroup$ – Breakfastisready Jan 10 at 22:38
1
$\begingroup$

Sorry for the sockpuppeting.

If we prove $ \alpha_q^{\text{cyclic}}$ to be continuous, we solve the long-standing problem of "are there good cyclic codes?".

This is because $ \alpha_q^{\text{cyclic}}(0) = 1$ trivially and by continuity we would then have $ \alpha_q^{\text{cyclic}}(\delta)> 0 $ for some $0 < \delta < 1$. This proves good cyclic codes.

As far as if the original proof is by Manin, I would still like to know!

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.