Proofs of Two Conjectures on Ternary Weakly Regular Bent Functions
Tor Helleseth, Henk D. L. Hollmann, Alexander Kholosha, Zeying Wang, Qing Xiang · IEEE Transactions on Information Theory · 2009
In this paper, we study ternary monomial functions of the formf(x) = Trn(axd), wherexisin \BBF3nandTrn: \BBF3nrarr \BBF3is the absolute trace function. Using a lemma of Hou, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising in the 2006IEEE Transactions on Information Theorypaper (vol. 52, pp. 2018-2032, 2006) are weakly regular bent, thus settling a conjecture of Helleseth and Kholosha. We also prove that the Coulter-Matthews bent functions are weakly regular.