What kind of symmetric codes have a very large minimum distance?

Question

I am in search of symmetric error-correcting codes that have a very large minimum distance. A codebook $\mathcal{C}$ is called symmetric if $\bf{1}\in\mathcal{C}$. Now, if $[n,k,d]_2$ is a symmetric code, then I want $d\approx n/2$. Can anybody tell me the approximate candidate for such type of code?

Answer 1

It depends on exactly what you mean by $d\approx n/2.$

If you absolutely want $d= n/2$ then the code cannot be that large in cardinality.

A first order Reed Muller code will have minimum distance $d=n/2,$ for $n=2^k.$ Its codewords can also be represented as rows of the $2^k\times2^k$ Sylvester Hadamard Matrix and its complement, i.e., $$ \left[ \begin{array}{c} H \\ J\oplus H \end{array} \right] $$ where $J$ is the all $1$'s matrix. So the number of codewords is at most twice the length. By the Plotkin bound if $C$ is a binary code of length $n$ and minimum distance $d\geq n/2$ we have $$ |C|\leq 2n. $$ See Theorem 4 in these notes.