In this paper, we recall how to construct new QAMs from a known one, and present how used the ortho-derivative method to figure out which of our new functions fall into different CCZ-classes. Based on these results and on others on smaller fields, we make to conjectures: that the full list of quadratic APN functions on F28 could be obtained using the QAM approached (provided enormous computing power), and that the total number of CCZ-inequivalent APN functions may overcome 50000.
We found 5412 new quadartic APN on F28 with the QAM method, thus bringing the number of known CCZ-inequivalent APN functions on F28 to 26525. Unfortunately, none of these new functions are CCZ-equivalent to permutations. A (to the best of our knowledge) complete list of known quadratic APN functions, including our new ones, has been pushed to sboxU for ease of study by others.