What kind of Symmetric Binary Linear Codes can satisfy my given specifications? I am looking for a binary linear code $\mathcal{C}$ which is symmetric, meaning $\mathbf{1}\in\mathcal{C}$. Other than that, my other requirement for the code are: if $\mathcal{C}:=[n,k,d]_2$ is a symmetric code, where $n$ is the length of codewords, $k$ is the dimension of code, and $d$ is the minimum distance of codes, then I require the following:
(1). $n$ is large ( can be in the range of $500$ to $10,000$),
(2). $ \lfloor\frac{n}{2}\rfloor-8 \leq d \leq  \lfloor\frac{n}{2}\rfloor -2$,
(3). $2^k\geq 4n$.
I am imposing these restrictions on the code because this kind of code finds its application in compressed sensing.
Here is what I have tried:
I constructed the symmetric BCH code with specifications $\mathcal{C}:=[8191, 14, 4095]_2$. As you can see here, $d=\lfloor\frac{n}{2}\rfloor$ and $2^k=2n+2$, but I need $2^k\geq 4n$. So, I thought that if I decrease the  value of $d$, then I might get $2^k \geq 4n$. But the catch here is, if I decrease the $d$ from $4095$, I get the new $d$ as $3968$, which means $d=\lfloor \frac{n}{2}\rfloor - 127$. I found that, in this case $d$ can not assume value in between $4095$ and $3968$. So, this kind of code is not useful for me because I also need to ensure that $ \lfloor\frac{n}{2}\rfloor-8 \leq d \leq  \lfloor\frac{n}{2}\rfloor -2$.
So, here I am asking for your help. Can you tell me any if there exist any kind of code that fulfills my requirement? Any help or hint regarding this would be highly appreciated.
 A: OK, I believe you can get a code of the type you want but won't quite get $d\geq n/2-8,$ as below from the Kerdock code via a construction due to Kiermaier and Zwanzger here. That's paywalled but the paper is available online elsewhere in paper-sharing sites.

You will not be able to get these parameters from a linear code. I don't know any other construction that would give what you want, since this construction builds on the original Kerdock via gray map construction, and is better than the best known way of getting more codewords via $\mathbb{Z}_4.$ See the end of this answer for more details and history.
This code has $n=1988$ and $M=2^12$ which is only $\geq 2n$ codewords. Its weight enumerator indicates codewords of weight $0,992,1024,1120.$
If you translate the code by the all 1's vector to double the number of codewords to $\geq 4n$ the weights will be $1988,1988-992,1988-1024,1988-1120$ which is $1988,996,964,868.$ The crucial question is the distances. I believe that the formal duality should establish the distance properties you need, but I am not so familiar with this construction. Perhaps @JyrkiLahtonen can comment. He may know a better construction.
Background: The Kerdock code's relationship to Z_4-linear codes was first discovered by Sole in 1991, and its full weight distribution, in the form of a quaternary sequence family (Family A) was given in Boztas' PhD thesis (1990) and later in Boztas, Hammons Kumar (1992), before the better known reference of the 5 authors paper which won the IT Transactions paper award in 1994. See the discussion in the Encyclopeadia of Mathematics online here which has references to those papers.
