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2.8. Other works 852.8 Other worksWe now compare our results with those of Cusick, Li and Stănică [62], and those of Carlet [37].2.8.1 Cusick et al.Cusick, Li and Stănică [62] proved the conjecture in some specific cases:• w H (t) = 1, 2,• t = 2 k − t ′ with w H (t ′ ) ≤ 2 and t ′ even,• t = 2 k − t ′ with w H (t ′ ) ≤ 4 and t ′ odd,by splitting each case in several subcases and using specific counting arguments for each one. Wenow compare their results with ours.The first case is treated by different theorems:• w H (t) = 1 if and only if t ≃ 1, so this case is taken care of by Theorem 2.4.8.• w H (t) = 2 if and only if t ≃ 3 which is included in Theorem 2.4.8 or d = 2 and α 1 = α 2 = 1which is included in the corollary of Theorem 2.4.15.The second one reads t = 2 k − t ′ = 1 + t ′ with w H (t ′ ) ≤ 2 and t ′ even. If w H (t ′ ) = 0, thent = 1. If w H (t ′ ) = 1, then t = 2 k − 2 i is made of one block which is included in Theorem 2.4.8. Ifw H (t ′ ) = 2, then our theorems can not be used to conclude.The last one reads t = 2 k − t ′ = 1 + t ′ with w H (t ′ ) ≤ 4 and t ′ odd, i.e. t = 0 or w H (t) ≥ k − 3(and t = 1 (mod 2) which is not important). If w H (t) = k − 1, then t ≃ −1. If w H (t) = k − 2,then t ≃ −3 which is made of one block and is included in Theorem 2.4.8, or t is made of twoblocks with β 1 = β 2 = 1 which is included in Theorem 2.4.15. The only subcases not directlyincluded in our Theorems 2.4.8, 2.4.14 and 2.4.15 when w H (t) = k − 3 are:• if d = 2:– 10010, but it is taken care of by the corollary of Theorem 2.4.15,– 001101 and 110010, which can be directly computed;• if d = 3:– 101010, but it is taken care of by Theorem 2.3.14,– one or two, but not three, α i ’s equal to 1, which is not treated by our theorems.Their approach kind of lacks a general strategy to tackle the conjecture, but points out therelevance of what we denote by r(a, t), the number of carries.2.8.2 CarletCarlet [37] proved the conjecture in the following cases:• w H (t) = 0, 1;• and t = 2 i − 2 j ;using affine functions and multisets. Both these results deal with numbers made of one block andare included in Theorem 2.4.8.