Affine Equivalency and Nonlinearity Preserving Bijective Mappings over 𝔽2

We first give a proof of an isomorphism between the group of affine equivalent maps and the automorphism group of Sylvester Hada-mard matrices. Secondly, we prove the existence of new nonlinearity preserving bijective mappings without explicit construction. Continuing the study of the group of nonlinearity preserving bijective mappings acting on \(n\)-variable Boolean functions, we further give the exact number of those mappings for \(n\,\le \,6\). Moreover, we observe that it is more beneficial to study the automorphism group of bijective mappings as a subgroup of the symmetric group of the \(2^{n}\) dimensional \(\mathbb {F}_{2}\)-vector space due to the existence of non-affine mapping classes.

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