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I am currently completing a Combinatorics homework and came across this question:

"A $q$-ary $[n, k, d]$ code is a subset $C$ of $\mathbb{F}^n_q$ ..."

What is the set $\mathbb{F}^n_q$?

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  • $\begingroup$ I know that that $\mathbb{F}$ is the notation for a field, but don't understand the use of superscript and subscript. $\endgroup$ – user148533 Feb 28 '15 at 15:53
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    $\begingroup$ \mathbb F_q is the field with q elements, so q is the number of symbols in your code (q=2 is quite common and usual). The superscript n is the dimension of the vector space in which the code sits; it is therefore the length of your code (number of symbols per codeword). $\endgroup$ – John Brevik Feb 28 '15 at 15:54
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$\mathbb{F}_q$ is the field with $q$ elements.
$q$ will be the power of a prime $q=p^k$. $\mathbb{F}_q$ can be thought of as polynomials of degree less than $k$, with coefficients in $\mathbb{Z}_p$ (integers modulo $p$);
where the polynomials are modulo $g(x)$ for some polynomial $g(x)$. $g(x)$ is irreducible and has degree $k$.
$\mathbb{F}_q^n$ is then vectors $(f_1,f_2,\ldots,f_n)$ where each $f_i$ is in the field $\mathbb{F}_q$.

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  • $\begingroup$ Ah thank you, that makes sense now, thank you $\endgroup$ – user148533 Feb 28 '15 at 15:57
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    $\begingroup$ @user148533 to thank you may want to upvote or accept the answer as it is the usual way of thanking here. $\endgroup$ – Seyhmus Güngören Feb 28 '15 at 16:02

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