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Classification of self dual quadratic bent functions. (English) Zbl 1264.06021
Every quadratic function from \(\mathbb F_2^t\) to \(\mathbb F_2\) is of the form \(f(x) = xAx^T+c\) for a unique constant \(c\) and a \(t\times t\) matrix over \(\mathbb F_2\) which is unique modulo \(\Lambda_t(\mathbb F_2)\), the group of the symmetric matrices with all diagonal entries \(0\). The associate alternating matrix of \(f\) is \(A+A^T\). The author shows that the quadratic function \(f(x) = xAx^T+c\) on \(\mathbb F_2^{2n}\) is a self-dual or anti-self-dual bent function if and only if \((A+A^T)^2 = I\) and \((A+A^T)A(A+A^T)+A^T \in \Lambda_{2n}(\mathbb F_2)\). This extends a result of C. Carlet et al. [Int. J. Inf. Coding Theory 1, No. 4, 384–399 (2010; Zbl 1204.94118)].
Using this result and the fact that (anti) self-duality is invariant under orthogonal coordinate transformations, the author completely classifies all self-dual and anti-self-dual quadratic Boolean bent functions in \(2n\) variables under the action of the orthogonal group \(O(2n,\mathbb F_2)\), by classifying all \(2n\times 2n\) involutionary matrices over \(\mathbb F_2\) under the action of \(O(2n,\mathbb F_2)\). The sizes of the \(O(2n,\mathbb F_2)\)-orbits of self-dual and anti-self-dual quadratic bent functions are determined explicitly.

06E30 Boolean functions
11E57 Classical groups
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
Full Text: DOI
[1] Carlet C., Danielsen L.E., Parker M.G., SolĂ© P.: Self dual bent functions. Int. J. Inform. Coding Theory. 1, 384–399 (2010) · Zbl 1204.94118 · doi:10.1504/IJICOT.2010.032864
[2] Dickson L.E.: Linear groups: with an exposition of the Galois field theory. Dover, New York (1958) · Zbl 0082.24901
[3] Green J.A.: The characters of the finite general linear groups. Trans. Amer. Math. Soc. 80, 402–447 (1955) · Zbl 0068.25605 · doi:10.1090/S0002-9947-1955-0072878-2
[4] Horn R.A., Johnson C.R.: Topics in matrix analysis. Cambridge University Press, Cambridge (1991) · Zbl 0729.15001
[5] Hou X.: GL(m, 2) acting on R(r, m)/R(r 1, m). Discret. Math. 149, 99–122 (1996) · Zbl 0852.94020 · doi:10.1016/0012-365X(94)00342-G
[6] Hou X.: On the asymptotic number of inequivalent binary self-dual codes. J. Combin. Theory Ser. A 114, 522–544 (2007) · Zbl 1145.94015 · doi:10.1016/j.jcta.2006.07.003
[7] Humphreys J.F.: A course in group theory. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1996). · Zbl 0843.20001
[8] Janusz G.J.: Parametrization of self-dual codes by orthogonal matrices. Finite Fields Appl. 13, 450–491 (2007) · Zbl 1138.94389 · doi:10.1016/j.ffa.2006.05.001
[9] Lidl R., Niederreiter H.: Finite fields. Cambridge University Press, Cambridge (1997) · Zbl 1139.11053
[10] Macdonald I.G.: Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1979). · Zbl 0487.20007
[11] MacWilliams J.: Orthogonal matrices over finite fields. Am. Math. Monthly 76, 152–164 (1969) · Zbl 0186.33702 · doi:10.2307/2317262
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