Orthomorphisms

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.
  • The situation for c=1 and c=2 was well known
  • index c=3
  • index c=4
  • index c=5
  • index c=6
  • index c=7
  • index c=8
  • index c=9
  • index c=10

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