On the invariance of perfect balancedness property under the choice of tapping sequence

The property of perfect balancedness of $k$-valued functions is of primary importance for cryptographic properties of stream ciphers constructed with such functions. The problem of description of the set of perfectly balanced $k$-valued functions for which the property of perfect balancedness is preserved for any choice of tapping sequence is considered. For the case of $2$-valued (Boolean) functions this problem was raised in 1996 by Golić and later addressed and fully solved in 2009 by the author. Recently we obtained some results for the case of $k$-valued functions. A classification of the known results on this subject (both obtained earlier and new) is provided, open problems and questions are formulated.