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

Let k ≥ 2 be an integer, and define S
_t
:= {(a, b) ∈ Z
²
|0 ≤ a, b ≤ 2k - 2, a + b = t(mod 2
^k
- 1), w(a) + w(b) ≤ k - 1}, where t ∈ Z, 1 ≤ t ≤ 2
^k
- 2. This paper gives the upper bound of the cardinality of S
_t
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.