One Note About the Tu-Deng Conjecture in Case $\mathop{\mathrm{w}}\nolimits(t)=5$

Let <inline-formula> <tex-math notation="LaTeX">$k \ge 2$ </tex-math></inline-formula> be an integer, and define <inline-formula> <tex-math notation="LaTeX">$S_{t}:=\{(a,b)\in \mathbb {Z}^{2} | 0 \le a,b \le 2^{k}-2, a+b=t (\text {mod} \,\,2^{k}-1), \mathop {\mathrm {w}}\nolimits (a)+ \mathop {\mathrm {w}}\nolimits (b)\le {k-1}\}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$t \in \mathbb {Z}, 1 \le t \le 2^{k}-2$ </tex-math></inline-formula>. This paper gives the upper bound of the cardinality of <inline-formula> <tex-math notation="LaTeX">$S_{t}$ </tex-math></inline-formula> in the case of <inline-formula> <tex-math notation="LaTeX">$\mathop {\mathrm {w}}\nolimits (t)=5$ </tex-math></inline-formula>. With this one, we conclude that a conjecture proposed by Tu and Deng in 2011 is right when <inline-formula> <tex-math notation="LaTeX">$\mathop {\mathrm {w}}\nolimits (t)=5$ </tex-math></inline-formula>.