In a very recent work by Tang and Maitra (IEEE Ttans Inf Theory 64(1):393–402, 2018], a theoretical construction of balanced functions f on n-variables (n≡2mod4\documentclass[12pt]{minimal}
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\begin{document}$$n\equiv 2 \bmod 4$$\end{document}) with very good autocorrelation and Walsh spectra values (Δf<2n2\documentclass[12pt]{minimal}
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\begin{document}$$\varDelta _f < 2^{\frac{n}{2}}$$\end{document} and nl(f)>2n-1-2n2+2n2-3-5·2n-24\documentclass[12pt]{minimal}
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\begin{document}$$nl(f) > 2^{n-1} - 2^{\frac{n}{2}} + 2^{\frac{n}{2}-3} - 5\cdot 2^{\frac{n-2}{4}}$$\end{document}) has been presented. The theoretical bounds could be satisfied for all such n≥46\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 46$$\end{document}. The case for n≡0mod4\documentclass[12pt]{minimal}
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\begin{document}$$n \equiv 0 \bmod 4$$\end{document} could not be solved in the said paper and it has also been pointed out that though theoretically not proved, such constructions may provide further interesting examples of Boolean functions. In this follow-up work, we concentrate in two directions. First we present a construction method for balanced functions f on n-variables (n≡0mod4\documentclass[12pt]{minimal}
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\begin{document}$$n\equiv 0 \bmod 4$$\end{document} and n≥52\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 52$$\end{document}) with Δf<2n2\documentclass[12pt]{minimal}
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\begin{document}$$\varDelta _f < 2^{\frac{n}{2}}$$\end{document} and nl(f)>2n-1-2n2\documentclass[12pt]{minimal}
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\begin{document}$$nl(f) > 2^{n-1} - 2^{\frac{n}{2}}$$\end{document}). Secondly, we apply search methods in suitable places to obtain balanced functions on even variables in the interval [10,…,26]\documentclass[12pt]{minimal}
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\begin{document}$$[10, \ldots , 26]$$\end{document} with improved parameters that could never be achieved before. As a consequence, for the first time we could provide examples of balanced Boolean functions f having Δf<2n2\documentclass[12pt]{minimal}
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\begin{document}$$\varDelta _f < 2^{\frac{n}{2}}$$\end{document} for n≡0mod4\documentclass[12pt]{minimal}
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\begin{document}$$n \equiv 0 \bmod 4$$\end{document}, where n=12,16,20,\documentclass[12pt]{minimal}
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\begin{document}$$n = 12, 16, 20,$$\end{document} and 24. Whatever functions we present in this paper have nonlinearity greater than 2n-1-2n2\documentclass[12pt]{minimal}
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\begin{document}$$2^{n-1} - 2^{\frac{n}{2}}$$\end{document}.
