By Charles Small

Textual content for a one-semester direction on the complex undergraduate/beginning graduate point, or reference for algebraists and mathematicians attracted to algebra, algebraic geometry, and quantity thought, examines counting or estimating numbers of recommendations of equations in finite fields focusing on to

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Comput. 9 (2), pp. 139–152, 1998. 58. H. Dobbertin. Almost perfect nonlinear power functions over GF (2n ): the Niho case. Inform. , 151, pp. 57–72, 1999. 59. H. Dobbertin. Almost perfect nonlinear power functions over GF (2n ): the Welch case. IEEE Trans. Inform. Theory, 45, pp. 1271–1275, 1999. 60. H. Dobbertin. Almost perfect nonlinear power functions over GF (2n ): a new case for n divisible by 5. Proceedings of Finite Fields and Applications FQ5, pp. 113–121, 2000. 61. H. Dobbertin. Private communication.

We have the following known facts on connection between CCZ-equivalence, isotopisms and strong isotopisms: • two planar DO polynomials F and F are CCZ-equivalent if and only if the corresponding commutative semifields SF and SF are strongly isotopic [22]; • two commutative presemifields of order p n with n odd are isotopic if and only if they are strongly isotopic [44]; • any commutative presemifield of odd order can generate at most two CCZequivalence classes of planar DO polynomials [44]; • if S1 and S2 are isotopic commutative semifields of characteristic p with the order of the middle nuclei and nuclei p m and p k , respectively, then one of the following statements must hold (a) m/k is odd and S1 and S2 are strongly isotopic, (b) m/k is even and either S1 and S2 are strongly isotopic or the only isotopisms between S1 and S2 are of the form (α N, N , L) where α is a non-square element of Nm (S1 ) [44] ; • there exist two commutative semifields of order 36 which are isotopic but not strongly isotopic [111].

Codes Cryptography 66(1–3), pp. 99–109, 2013. 102. A. Pott, Y. Tan, T. Feng, and S. Ling. Association schemes arising from bent functions. Des. , vol. 59, no. 1–3, pp. 319–331, Apr. 2011. 103. O. S. Rothaus. On “bent” functions. J. Combin. Theory Ser. A, vol. 20, no. 3, pp. 300–305, 1976. 104. V. Sidelnikov. On mutual correlation of sequences. Soviet Math. , 12, pp. 197–201, 1971. 105. Y. Tan, A. Pott, and T. Feng. Strongly regular graphs associated with ternary bent functions. J. Combin. Theory Ser.