MOS substitution: Difference between revisions

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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.

Case 2: neither of ''k'' and ''k + 1'' equals (''b'' + ''c''). Here, if '''Y''' occurs ''j'' or ''j'' + 1 times in a window of size ''k'', then ''Y'' occurs ''j'' + 1 or ''j'' + 2 times in a window of size ''k'' + 2.

Case 2: neither of ''k'' and ''k'' + 1 equals (''b'' + ''c''). Here, if '''Y''' occurs ''j'' or ''j'' + 1 times in a window of size ''k'', then ''Y'' occurs ''j'' + 1 or ''j'' + 2 times in a window of size ''k'' + 2.

== Open questions ==

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 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.
-Case 2: neither of ''k'' and ''k + 1'' equals (''b'' + ''c''). Here, if '''Y''' occurs ''j'' or ''j'' + 1 times in a window of size ''k'', then ''Y'' occurs ''j'' + 1 or ''j'' + 2 times in a window of size ''k'' + 2.
+Case 2: neither of ''k'' and ''k'' + 1 equals (''b'' + ''c''). Here, if '''Y''' occurs ''j'' or ''j'' + 1 times in a window of size ''k'', then ''Y'' occurs ''j'' + 1 or ''j'' + 2 times in a window of size ''k'' + 2.
 == Open questions ==