Relative distance in Codes I'm studying coding theory. In my lecture we saw that Hadamard codes have an optimal relative distance of $1/2$. The relative distance of a code $C$ with minimum distance $d(C)$ and block length $n$ is defined by:
$$\dfrac{d(C)}{n}.$$
According to this, the minimum distance for a Hadamard code with block length $n=2^k$ is $\dfrac{n}{2} = 2^{k-1}$. Is there any code $C$ with block length $n$ and minimum distance $d(C)>n/2$? How can I prove it?

It depends what kind of code you want. In its simplest form, a $q$-ary block code of length $n$ is just a subset of $\mathbf{F}_q^n$. Hence it is trivial to construct a block code of length $n$ with minimal distance $n$: $\{0^n,1^n\}$. This code is also linear, and even cylic.
