MOS substitution: Difference between revisions
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=== MOS substitution scales have block balance at most 2 === | === MOS substitution scales have block balance at most 2 === | ||
Consider a MOS substitution scale a'''X''' (b'''Y''' c'''Z'''). It is obvious that '''X''' has block balance 1, since we can replace the MOS substitution scale with the MOS scale a'''X''' (b + c)'''W''' to make this argument. '''Y''' and '''Z''' have block balance at most 2, since we can consider windows of the MOS scale of size ''k'' or ''k'' + 1, and the number of times '''Y''' (and also '''Z''') differs by at most 2. This is proved below for '''Y''', but it's exactly the same argument for '''Z''': | Consider a MOS substitution scale a'''X''' (b'''Y''' c'''Z'''). It is obvious that '''X''' has [[block balance]] 1, since we can replace the MOS substitution scale with the MOS scale a'''X''' (b + c)'''W''' to make this argument. '''Y''' and '''Z''' have block balance at most 2, since we can consider windows of the MOS scale of size ''k'' or ''k'' + 1, and the number of times '''Y''' (and also '''Z''') differs by at most 2. This is proved below for '''Y''', but it's exactly the same argument for '''Z''': | ||
Case 1: one of ''k'' and ''k'' + 1 equals (''b'' + ''c'') and '''Y''' occurs exactly ''b'' times or ''b'' plus or minus 1 in this case. | Case 1: one of ''k'' and ''k'' + 1 equals (''b'' + ''c'') and '''Y''' occurs exactly ''b'' times or ''b'' plus or minus 1 in this case. | ||