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Yes!!! Sorry I'm nuts! I was missing a digit at the beginning of the second number AND I was applying an XNOR operator. The right one should be: 010101 XOR 011000 = 001101 Sorry!

Thank you vey much for your answer. Well I'm, trying to understand: we have a cyclic series of 64 different 6-uplets and another series of 6-uplets (let's call them switches) that result form XORing two adjacent 6-uplets of the first series. If I apply to a starting arbitrary 6-uplet a whole series of switches, at some stage I'll have a repeating 6-uplet (corresponding to a 000000 switch). So what I look for is impossible for 64 different 6-uplets AND 64 different switches. Is it? So next question is: what about 64 different 6-uplets and a set of 64-switches that don't include the 000000 switch, but instead the repetition of a another (arbitrary) switch? Did I understand it correctly?

I am not a mathematician, but rather a musician who met a problem (it's not homework!) that can possibly be posed as a mathematical problem. So I'm looking for a solution but also and especially for a direction to the discipline that may solve the problem. I apologize for bad use of proper mathematical writing. Let's say I have the whole set of all 64 possible 6-tuplets of digital bits, such as 000000, 000001, (000010) and so on up to 111111. Of course I can operate on couples of 6-uplets in order to have the XOR result. For example: 010101 XOR 11000 = 011010. I would like to put the 64 6-uplets in such an order (a cyclic one with period 64, that is to say in circle) so that of course these 64 6-uplets appear only once but, and that's what is difficult for me, in such a way so that also the XOR results of two adjacent 6-uplets appear to be in such an order so that each XOR result is present only once (a part form the 000000 that of course is not present and another 6-uplet that is present twice, in order to have the cycle of 64 6-uplets resulting from the XOR operation corresponding to the starting cycle of 64 6-uplets). I apologize for bad explaining, I hope I was clear. I would like to know if such a (double) order of 6-uplets is possible and under which conditions. Of course I would like to understand what discipline is best fitting this problem... Group Theory? Lattice Theory? Thank you for any help!!!