文摘
Let k be a positive integer. For any positive integer x = Σ i=0 ?/sup> x i 2 i , where x i = 0, 1, we define the weight w(x) of x by w(x) ?Σ i=0 ?/sup> x i . For any integer t with 0 < t < 2 k ?1, let S t ?{(a, b) ??sup>2|a + b ?t (mod 2 k ?1), w(a) + w(b) < k, 0 ?a, b ?2 k ?2}. This paper gives explicit formulas for cardinality of S t in the cases of w(t) ?3 and an upper bound for cardinality of S t 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.