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In 2006, 9-variable Boolean functions having nonlinearity 241, which isstrictly greater than the bent concatenation bound of 240, have been discoveredin the class of Rotation Symmetric Boolean Functions (RSBFs) by Kavut, Maitraand Yucel. To improve this nonlinearity result, we have firstly defined somesubsets of the n-variable Boolean functions as the "generalized classes ofk-RSBFs and k-DSBFs (k-Dihedral Symmetric Boolean Functions)", where k is apositive integer dividing n and k-RSBFs is a subset of l-RSBFs if k < l.Secondly, utilizing the steepest-descent like iterative heuristic searchalgorithm used previously to identify the 9-variable RSBFs with nonlinearity241, we have made a search within the classes of 3-RSBFs and 3-DSBFs. Thesearch has accomplished to find 9-variable Boolean functions with nonlinearity242 in both of these classes. It should be emphasized that although the classof 3-RSBFs contains functions with nonlinearity 242; 1-RSBFs or simply RSBFs,which is a subset of 3-RSBFs, does not contain any. This result also shows thatthe covering radius of the first order Reed-Muller code R(1, 9) is at leastequal to 242. Thirdly, motivated by the fact that RSBFs are invariant under aspecial permutation of the input vector, we have classified all possiblepermutations up to the linear equivalence of Boolean functions that areinvariant under those permutations. Specifically, for 9-variable Booleanfunctions, 9! possible permutations are classified into 30 classes; and thesearch algorithm identifies some of these classes as "rich". The rich classesyield new Boolean functions with nonlinearity 242 having differentautocorrelation spectra from those of the functions found in the generalized3-RSBF and 3-DSBF classes.

9-variable Boolean Functions with Nonlinearity 242 in the Generalized Rotation Class