Necklaces and Slimes
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It was asked if one can find a bijective map between the following two objects: binary necklaces with n black beads and k white beads and certain (n, k)-codes whose weighted sum is 0 modulo n (Brauner et al.,2019 ; Chan, 2019 ). The former object is one that has been studied for ages, whereas the latter one was shown to be the states in a dollar game played on a cyclic graph (Corry & Perkinson, 2018 ).
The case when n and k are coprime was shown in  and it is easily described by using rotation. We show that in the general case, all that one needs to construct the bijective map is to construct a rotation-invariant and weight increasing map (riwi-map) on the codes. When n and k are coprime the simple cyclic rotation works as a riwi-map. We show that when n or k is prime, a new map called a slime migration works as a riwi-map and hence allows one to get a bijective map as a result.