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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)
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