A note on the Tu-Deng conjecture

DOI: 10.1007/s11424-015-2240-3 Received: 8 November 2012 / Revised: 1 July 2014 c ©The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2015 Abstract Let k be a positive integer. For any positive integer x = ∑∞ i=0 xi2 , where xi = 0, 1, we define the weight w(x) of x by w(x) := ∑∞ i=0 xi. For any integer t with 0 < t < 2 k − 1, let St := {(a, b) ∈ Z2|a+ b ≡ t (mod 2 − 1), w(a)+w(b) < k, 0 ≤ a, b ≤ 2 − 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.