# What are dual codes and the codewords denoted by these dual codes in terms of trace?

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

• 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. – kodlu Feb 10 '19 at 10:56