Asymptotic for binary linear code

Question

Let $C=[n,k,d]$ is a binary linear code of length $n$, dimension $k$, and minimum distance $d$. Let us consider all possible binary linear codes with $k=d$ and $n\in \mathbb N$. Is it true that $$A=\mathop{\lim }\limits_{k\to \infty } \frac{k}{n}=0\ ?$$ I look codetables.de and this assumption looks plausible. Are suitable boundaries or hypotheses known?

Answer 1

The Gilbert Varshamov bound shows the existence of a binary linear code of dimension $$k\geq n-\lfloor \log_2 V(n,d-1)\rfloor,$$ where $V(n,r)$ is the volume of the hamming sphere in the $n$ dimensional hypercube.

Let $d=k$, and find the maximal $k$ satisfying this equation, for each $n$. Along a subsequence of integers, something like the existence of codes with parameters $$ [Ak_i,k_i,k_i] $$ with $A=5,$ should hold.

More generally, let $n\rightarrow \infty$ and use the fact that for each $n$ the binomial coefficients $\binom{n}{k},$ are superincreasing in $k$ up to about $k<n/3,$ and the approximation $$ V(n,k-1)\approx \binom{n}{k} \approx 2^{n (h(k/n)+o(1))}, $$ where $h$ is binary Shannon entropy (see here) to conclude that the solution $\theta=0.227\cdots$ to $$ 1-\theta=h(\theta) $$ where $\theta=k/n,$ means that asymptotically there is a sequence of codes with parameters $[Ak,k,k]$ and $A=1/\theta.$