Finite Fields and Their Applications, 91, (October 2023).
doi: 10.1016/j.ffa.2023.102257
Abstract:
Let S be a dimensional dual hyperoval of rank n over GF(2). We introduce and study the radical P(S), which is a subspace of the ambient space U(S) of S invariant under the automorphism group of S. For the vast majority of the known dimensional dual hyperovals we have P(S)=U(S). Interesting is the case of proper radicals, i.e. P(S) ≠ U(S). Starting point of our investigations is a result of the second author [10, Thm. 1], [7, Thm. 3.6] (Theorem 1.2 below) which characterizes alternating dual hyperovals by the property that S splits over P(S). This Theorem is extended by Theorem 1.3 where we characterize dimensional dual hyperovals S with dim U(S) - dim P(S) = rank(S) -1. Moreover we will show (Theorem 4.6) that a proper radical implies that this dimensional dual hyperoval is a disjoint union of subDHOs of smaller rank. The notion of ''disjoint union of subDHOs'' has been introduced by Yoshiara [17]. Some theory on dimensional dual hyperovals with proper radicals is developed. Our paper also provides some computational results on dual hyperovals of small rank with a proper radical. These calculations indicate - though dual hyperovals with a proper radical seem to be scarce - that the number of these hyperovals is steadily growing as function of the rank.
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Additional resources for the paper:
The program used for the proof in Section 6.2,
Details for Section 7.
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