On some Representations of Quadratic APN Functions and Dimensional Dual Hyperovals

RIMS Kôkyûroku, 1687 (2010), 118-130.


APN functions have become a quite popular area of research and especially quadratic APN functions have links to various other mathematical areas. One link is to geometry, namely dimensional dual hyperovals. Quadratic APN functions are equivalent to certain subspaces of the alternating bilinear forms, and most examples of dual hyperoval are, or can be, constructed from certain subspaces of the bilinear forms. As consequence different related publications use substantially different notation. The present manuscript is based on the authors efforts to provide an explicit translation of the notation found in several papers on APN functions and dual hyperovals by Nakagawa, Taniguchi and Yoshiara, into his own notation and back.
The manuscript contains also an alternative proof of Taniguchi's result [1,Theorem 11].

[1] H. Taniguchi. On the duals of certain d-dimensional dual hyperovals in PG(2d+1, 2). Finite Fields And Their Applications, 15:673-681, 2009.

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