A question about a possible algebraic solution

[mvl]

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!!!

[uncool]

Well, for one thing, it's impossible directly, using some linear algebra over the field of 2 elements.

 

Basically, assume that 63 of them are the distinct nonzero switches (i.e. 000001, 000010, ..., 111111); if it's cyclic, we should be able to figure out what the last element is. We'll check the 6th bit first; 31 of the switches don't change it, while 32 do, so in total, it hasn't been switched. The same reasoning works for each of the bits - so in the end, none of the bits have changed. That means that the 64th switch must be the 0 switch - 000000. In other words, you'll get two that are the same. This happens even if you don't require it to be cyclic.

 

The next question, then, would be whether it's possible to do 62 of them, and have both the outcomes and the switches be different.

=Uncool-

Edited July 27, 2013 by uncool

[mvl]

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?

[pears]

Of course I can operate on couples of 6-uplets in order to have the XOR result. For example: 010101 XOR 11000 = 011010.

 

Shouldn't that be 001101? Sorry to be such a pedant but it's driving me crazy. Or am I wrong? I'm tired and really ought to go to sleep.

[mvl]

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!

[pears]

Thank you for putting me out of my misery

[uncool]

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).

Not quite. What I'm saying is there is only one set of 63 different switches, and if you apply all of them, you'll have effectively done nothing.

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?

There would be 63 different switches in a row somewhere - so you get the same problem.

=Uncool-