I am currently reading a research paper (linked below) that mentions "Consider only maps that vanish at 0, their short codes $C_f$ and their duals $C_f^\perp$. The duals can be written as $C_f^\perp = \langle X \rangle \oplus \langle f(x) \rangle$ where for any map $g:V \rightarrow V$, we use notation $\langle g(x) \rangle$ to denote code whose codewords are $[Tr(ag(\omega^t):t \in [0 \dots 2^{m}-2]]$, $a \in V$. Here $Tr$ denotes the absolute trace from $GF(2^m)$ to $GF(2)$."

I am having trouble understanding why dual codes can be written as $C_f^\perp = \langle X \rangle \oplus \langle f(x) \rangle$ and how codewords can be denoted by trace of the expression above. It would be great if anyone could explain what "absolute trace from $GF(2^m)$ to $GF(2)$." means.

An APN Permutation in Dimension 6

  • $\begingroup$ I think you need to read a book on algebraic coding theory. Jones and Jones have a nice book in springer universitext series. This question is a math question on coding theory, so ask in math.stackexchange and delete here if you like. $\endgroup$ – kodlu Feb 10 at 10:56

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