A Note on the Five Valued Conjectures of Johansen, Helleseth and Kholosha and Zeta Functions

Abstract

For the complete five-valued cross-correlation distribution between two m-sequences s t and s dt of period 2 m − 1 that differ by the decimation d = (2 2k + 1)/(2 k + 1) where m is odd and gcd(k, m) = 1, Johansen and Helleseth expressed it in terms of some exponential sums. And two conjectures were presented about them. In this correspondence we study these conjectures for the particular case where k = 3, and the cases k = 1, 2 can also be analyzed in a similar process. When k > 3, the degrees of the relevant polynomials will become higher.

Three Classes of Optimal Ternary Cyclic Codes and the Weight Distributions of Their Duals

“…Since cyclic codes have efficient encoding and decoding algorithms, they have been widely used in communication systems, consumer electronics and data storage systems. Also, they have applications in cryptography [2] and sequence design [5,21] . So, a number of scholars have been encouraged to study cyclic codes over the last few decades [1,[6][7][8][9][10][11][12][13][14][15][16][17][18] .…”