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Problem 8: Distinct Periodic Words

In this problem we examine how much different two periodic words of the same length can be. The difference is measured with the Hamming distance. The Hamming distance between $x$ and $y$ of the same length is $\HAM(x,y)=|\{j \mid x[j]\neq y[j]\}|$.

We consider a word $x$ whose period is $p$, a word $y$ of length $|x|$ whose period $q$ satisfies $q\leq p$ and we assume there is at least a mismatch between them. Let $i$ be the position on $x$ and on $y$ of a mismatch, say, $x[i]=\sa{a}$ and $y[i]=\sa{b}$. In the picture $x=u^2$, $|u|=p$, and $|v|=q$.

Distinct Periodic Words

What is the minimal Hamming distance between two distinct periodic words of the same length?
Consider different cases of position $i$ according to periods $p$ and $q$.


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  • A. Amir, C. S. Iliopoulos, and J. Radoszewski. Two strings at Hamming distance 1 cannot be both quasiperiodic. Inf. Process. Lett., 128:54-57, 2017.