One Note About the Tu-Deng Conjecture in Case <inline-formula> <tex-math notation="LaTeX">$\mathop{\mathrm{w}}\nolimits(t)=5$ </tex-math></inline-formula>

Abstract

Let k ≥ 2 be an integer, and define St := {(a, b) ∈ Z2|0 ≤ a, b ≤ 2k - 2, a + b = t(mod 2k - 1), w(a) + w(b) ≤ k - 1}, where t ∈ Z, 1 ≤ t ≤ 2k - 2. This paper gives the upper bound of the cardinality of St in the case of w(t) = 5. With this one, we conclude that a conjecture proposed by Tu and Deng in 2011 is right when w(t) = 5.

Keywords

• Tu-Deng conjecture
• algebraic immunity
• Boolean function
• Hamming weight