I have the following definition:
Let $C$ be an $\mathbb {F} _{p}-[n, k, d]$ code. We say that $C$ is systematic on a set of $k$ coordinates $\{i_1, ..., i_k\}$ if there is exactly one codeword $c = (c_{i_{1}}, ..., c_{i_{k}}) \in \mathbb {F}_{p}^k$.
I am struggling to understand this definition - what does it mean explained in plain English? My understanding is that, for example, if $C$ is systematic on its first $k$ coordinates, then the first $k$ coordinates of each codeword are just the original message of length $k$.
Furthermore, what does it then mean to say that $C$ is systematic on every set of $k$ coordinates?
It means that for any $k$ coordinates out of $n,$ and there are $\binom{n}{k}$ of those, those coordinates could conceivably serve as the original message coordinates. So it's a strong equidistribution property.
In other words, take the code $C$ apply any permutation $\sigma$ in $S_n$ to its coordinates, to obtain a new code $C_{\sigma}$ via the map $$ (c_1,\ldots,c_n) \mapsto (c_{\sigma(1)},\ldots,c_{\sigma(n)}), $$ the new code $C_{\sigma}$ is still systematic, since on the coordinates $$ (\sigma^{-1}(1),\ldots,\sigma^{-1}(k)) $$ each $k-$tuple occurs exactly once.