"Manin's theorem" reference, coding theory

Question

Question:

In the book [1], a theorem called "Manin's Theorem" is stated on page 19

"There exists a continuous decreasing function $\alpha_q : [0,1] \rightarrow [0,1]$ such that

• $\alpha_q$ is strictly decreasing on $[ 0,\frac{q-1}{q}]$,
• $\alpha_q(0) = 1$,
• if $\frac{q-1}{q} \leq x \leq 1$, then $\alpha_q(x)=0$,
• $\Sigma_q = \lbrace (\delta,R) \in [0,1]^2 \mid 0 \leq R \leq \alpha_q(\delta) \rbrace$."

The citation for this is given as [2], but on Google Scholar this is some random book from 1903. It's also the case that "Manin's theorem" isn't a very specific name, considering the amount of varied work done by Manin. I'm looking for a place to find [2] online, or another alternate reference to find this theorem and its proof.

References:

[1] Joyner, David, and Jon-Lark Kim. Selected unsolved problems in coding theory. Springer Science & Business Media, 2011.

[2] Shokranian, Salahoddin, and Mohammad Amin Shokrollahi. Coding theory and bilinear complexity. Vol. 21., Juelich, International Office of the Forschungszentrum Juelich, 1993.

Answer 1

Here is the answer to the question as stated literally:

Citation 2 is found here and the paper of Manin is here, see in particular Theorem 3 on p. 716. Still, I don't see the exact correlation to what the authors Joyner and Kim write in their book.

Answer 2

My impression is that the intention of the authors is to introduce the function $\alpha_q$ and to state a few of its properties. But the way they do it ("there exists ... such that") looks a bit clumsy to me. Why not first give the common definition of $\alpha_q$ and then state its properties.

The function $\alpha_q(\delta)$ is commonly defined as the best possible asymptotic information rate of a $q$-nary block code of asymptotic relative distance $\delta$.

See for example Section 2.10 in the book Fundamentals of Error-Correcting codes by W. Cary Huffman & Vera Pless, 2003. A lot of other textbooks written for a mathematical audience should do, too.

There are lower and upper bounds known for the function $\alpha_q$, but the exact values are unknown (in the open interval $(0,1 - \frac{1}{q})$). This is an important open problem in coding theory, most prominently in the binary case $q = 2$.

Let me remark that the term "Manin's theorem" is new to me. A few remarks on your four listed points:

• EDIT: Actually, I'm not sure about the property strictly decreasing on $[0,1-\frac{1}{q}]$. Has this really been shown? Where? Does anyone have a reference?
• The asympotic rate of the full Hamming space gives $\alpha_q(0) = 1$.
• This follows from the asymptotic Plotkin bound and the specific value $\alpha_q(1 - \frac{1}{q}) = 0$ follows from the asymptotic Elias-Bassalygo bound (or the knowledge that $\alpha_q$ is continuous).
• I guess, this is the common definition of $\alpha_q$ (I don't know what $\Sigma_q$ is).