Let $Ham$ be the $[7,4,3]_2$ Hamming code.
It is known that $\{w(c):c\in Ham\}\subseteq\{0,3,4,7\}$, where $w(c)$ is the Hamming weight of the word $c$.
A code $C$ of length $n$ is called cyclic if $\forall c=(c_0,\dots,c_{n-1})\in C:(c_{n-1},c_0,\dots,c_{n-2})\in C$.
A straight forward extension for $Ham$ to an $[8,4,4]_2$ code $C$ is a code which is restricted to the $7$ first coordinates is $Ham$ and the last (new) coordinate is a parity bit for the first (original) $7$ ones.
This is true since $C$ is of length $8$ which is $7$ for the first $Ham$ coordinates and another $1$ for the parity bit. Also we end up with the same number of codewords, so the dimension is still $4$. But, now $\{w(c):c\in C\}\subseteq\{0,4,8\}$.
The question is whether $C$ is a cyclic code? and a more general one is what are all the $[8,4]_2$ cyclic codes (I mean is there a certain criterion)?
I know that $Ham$ is cyclic.
I can also say that in $Ham$ we can look at the zero codeword, the all $1$'s codeword and two other codewords $x,y$ with all their cyclic movements this gives us $1+1+7+7=16$ codewords.