This data was produced by my student David Fear for his MSc thesis.
An orthomorphism of a finite field F is a permutation θ:F➞F
such that the map x➞θ(x)-x is also a permutation of F.
Suppose |F|=q and k divides q-1.
An orthomorphism θ is cyclotomic of index k if
θ(x)/x is constant on the cosets of the subgroup of index k in F*.
The value of θ(x)/x is known as a multiplier. To specify a cyclotomic
orthomorphism we may just list the multipliers. There will be 1 for a linear
orthomorphism, 2 for a quadratic, 3 for a cubic, 4 for a quartic and so on.
David needed the following data to support claims made in his thesis.
In every case q is the order of the field.
- There is a cyclotomic orthomorphism of index (q-1)/2
provided q is odd and in the range 9≤q<1000.
- There is a quadratic orthomorphism orthogonal to a quartic orthomorphism
in every field such that q=1 mod 4 and q≥13. His thesis gives a
theoretical proof for q>264192. Here we give examples that fill the gap
up to that bound. In most cases the quartic could be chosen to be nearlinear,
these examples can be found here.
For a few small fields such examples weren't available.
These cases are dealt with here.
- Let c be an integer, 1≤c≤10 and q=1 mod c. There exists
a linear orthomorphism orthogonal to an index c orthomorphism in a field
of order q if and only if (c,q) is not one of the pairs
(1, 2), (1, 3), (2, 3), (3, 4), (2, 5), (4, 5), (2, 7), (3, 7), (6, 7), (4, 13).
Again, this is known beyond a certain point. To fill the gap, the
following examples suffice. In each case the linear orthomorphism is
given first, then the index c orthomorphism. If the latter can be given
as a near-linear orthomorphism then it is (with just two multipliers
given). Otherwise, c multipliers are given.
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