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?

  • $\begingroup$ On a second thought, I think proving it to be continuous will solve the long-standing "good cyclic codes" conjecture. $\endgroup$ Jan 10, 2019 at 21:54

2 Answers 2


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.

  • $\begingroup$ 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) $\endgroup$ Jan 10, 2019 at 22:38

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!


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