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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.

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