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I want to determine all values of $q$ such that $C$ is an $[8,k,3]$ Hamming code over $\mathbb{F}_q$. Is there a way to do this that's not just computing every possible values of $q$? Since $k$ is fixed I thought there would be a way to use this, but I don't see it.

Saegusa
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  • Isn't that exactly the same as testing every possible value of $q$? Because I get $\frac{q^r-1}{q-1}=8$, and then I just solve for $q$. But I don't see how to do that unless we specify $r$. For instance, if $r=2$, we get $q=7$. Do I have to do this for all $r$? – Saegusa May 16 '21 at 20:13
  • Ahh didn't think of using this. Thanks you very much! – Saegusa May 17 '21 at 20:17

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