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APN Functions

We have used our program for the canonization of linear codes in order to check CCZ-inequivalence of APN-functions f: GF(2n) -> GF(2n) and to calculate their automorphism groups. The codes have length 2n and dimension 2n+1. (w denotes a primitve Element of GF(2n))

These functions were proposed by Carl Bracken, Eimear Byrne, Nadya Markin, Gary McGuire in their paper A few more quadratic APN functions and John Cannon


CCZ-inequivalent APN-functions, n=12:

Nr.

APN-function

Order of the automorphism group

Calculation time (in seconds)

1 x3

201277440

360

4 w16x768+wx33+x257+w290x544

28672

??? (modifications by hand necessary)

5 w16x768+wx33+w290x544

114688

63000

6 w16x768+wx33+x257

114688

62500

7 x993

49140

97500

 

The codes are all linearly nonisometric since the orders of the automorphism group are all different except for the codes 5 and 6. A closer look on the canonical generator matrix reveals that these codes are also inequivalent.

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