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APN polynomials and related codes. (English) Zbl 1269.94035
Summary: A map $$f: \mathrm{GF}(2^m)\to \mathrm{GF}(2^m)$$ is almost perfect nonlinear, abbreviated APN, if $$x\mapsto f(x+a)-f(x)$$ is 2-to-1 for all nonzero $$a$$ in $$\mathrm{GF}(2^m)$$. If $$f(0) =0$$, then this condition is euqivalent to the condition that the binary code of length $$2^m-1$$ with parity-check matrix
H:=\left[ \begin{align*}{\ldots &\, \;\quad \omega^j\quad\ldots\cr \ldots& \, \, \;f(\omega^j)\, \;\ldots}\end{align*}\right]
is double-error-correcting, where $$\omega$$ is primitive in $$\mathrm{GF}(2^m)$$.
We give a brief review of these maps and their polynomials; and we present some new examples along with some related codes and designs which serve as invariants for their equivalence classes.

##### MSC:
 94B05 Linear codes, general 11T06 Polynomials over finite fields 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)