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88 Chapter 2. On a conjecture about addition modulo 2 k − 1Recall that t (or an equivalent one) can be written down asα 1{β 1{α i{t = 1---10---0... 1---10---0... 1---10---0t 1 t iThen, each integer coefficient µ i of the polynomiali=0β i{α d{β d{t d.k−1∑ ()µ i x i = Tr B β d−10,0 B 0,1 B α d−11,1 B 1,0 · · · B β1−10,0 B 0,1 B α1−11,1 B 1,0= Tr ( B β dC αd · · · B β1 C α1)represents the number of closed paths in D with length k and weight i, i.e. the number of couples(a, b) ∈ ( Z/(2 k − 1)Z ) 2such that a + b = t and r(a, b) = i. Hence, we get the equality#S t,k =k−1−w H (t)∑To check the validity of the conjecture for a given t, it is therefore sufficient to compute the traceof a product of at most k (4, 4)-matrices.Moreover, it is a basic result that testing one t in each cyclotomic class is enough. For example,the smallest one, which is called the cyclotomic leader, can be chosen.To summarize, the algorithm devised by Tu and Deng to check the validity of their conjectureis described in Algorithm 2.1.i=0µ i .Algorithm 2.1: Tu–Deng algorithmInput: A positive integer k ≥ 2Output: True if the conjecture is verified for k, False otherwise1 Compute the set T of cyclotomic leaders modulo 2 k − 12 foreach t in T do3 Compute the sets {α i } and {β i } for t4 Compute the polynomial ∑ k−1i=0 µ ix i = Tr ( )B β dC αd · · · B β1 C α15 if ∑ k−1−w H (t)i=0µ i > 2 k−1 then6 return False7 return True2.9.2 Necklaces and Lyndon wordsThe link between cyclotomic equivalence modulo 2 k − 1 and rotation of binary strings shows thatthe generation of cyclotomic leaders modulo 2 k − 1 is nothing but the classical combinatorialproblem of generation of necklaces with k beads of up to two different colors.Definition 2.9.1 (Necklace, Lyndon word). Let k and n be two strictly positive integers.A k-ary necklace of length n, also called a necklace of n beads in k colors, is a string of lengthn on an alphabet of size k, up to rotation.An aperiodic necklace is called a Lyndon word.