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I am only familiar with binary implementations of information, though I am aware other implementations (ternary etc.) exist. I was wondering, if a code over $\mathbb{F}_q^n$ is used in practice, where $q\neq 2$, how is this done really? Do we use chunks of bits to encode elements of $\mathbb{F}_q$, for example $\{00, 01, 10, 11\} \in \mathbb{F}_4$? Or is a different protocol used? I cannot image it's the former since this would be hard for $q\neq 2^s$ for integer $s$, and it requires block encoding which is less robust against errors.

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For $\mathbb{F}_2^n$ people store the bits consisting of the coefficients of the polynomial of some generator $x$. For $\mathbb{F}_q$ the field is isomorphic to simple integers mod $q$, so they simply store the integer in binary. I don't know of anything that actually uses $\mathbb{F}_q^n$ with $q > 2, n \geq 2$, but again you could just store a vector of coefficients of some polynomial in generator $x$, where each coefficient is an integer $<q$.

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