A note on the Tu-Deng conjecture

Let k be a positive integer. For any positive integer x = Σi=0∞xi2i, where xi = 0, 1, we define the weight w(x) of x by w(x) ≔ Σi=0∞xi. For any integer t with 0 < t < 2k − 1, let St ≔ {(a, b) ∈ ℤ2|a + b ≡ t (mod 2k − 1), w(a) + w(b) < k, 0 ≤ a, b ≤ 2k − 2}. This paper gives explicit formulas for cardinality of St in the cases of w(t) ≤ 3 and an upper bound for cardinality of St when w(t) = 4. From this one then concludes that a conjecture proposed by Tu and Deng in 2011 is true if w(t) ≤ 4.