Correlation immunity of symmetric Boolean functions is studied in this paper. Lower bounds on the number of constructible correlation immune symmetric functions are given. Constructions for such new balanced functions are presented. These functions are also known as 1-resilient functions. In 1985, Chor et al. conjectured that the only 1-resilient symmetric functions are the exclusive-or of all n variables and its negation. This conjecture, however, was disproved by Gopalakrishnan, Hoffman and Stinson in 1993 by giving a class of infinite counterexamples, and they noted that it does not seem to extend any further in an obvious way. In this paper two more infinite classes of such examples are presented for n being even and being odd, respectively, and consequently one of the two open problems proposed by Gopalakrishnan et al., is addressed by constructing new symmetric resilient functions. © 2003 Elsevier B.V. All rights reserved.
Wu, C. K., & Dawson, E. (2004). Correlation immunity and resiliency of symmetric Boolean functions. Theoretical Computer Science, 312(2–3), 321–335. https://doi.org/10.1016/j.tcs.2003.09.009