Hamming Distance for Conjugates

TL;DR

This paper investigates the properties of Hamming distances between conjugate strings, revealing differences between binary and larger alphabets and establishing bounds and conditions for these distances.

Contribution

It introduces new bounds and conditions for the Hamming distance between conjugate strings over various alphabet sizes, expanding understanding of their combinatorial properties.

Findings

Over binary alphabets, the Hamming distance is always even.

For alphabets of size 3 or greater, the Hamming distance can range widely, excluding 1.

The smaller string can be assumed to have only one letter type in these analyses.

Abstract

Let x, y be strings of equal length. The Hamming distance h(x,y) between x and y is the number of positions in which x and y differ. If x is a cyclic shift of y, we say x and y are conjugates. We consider f(x,y), the Hamming distance between the conjugates xy and yx. Over a binary alphabet f(x,y) is always even, and must satisfy a further technical condition. By contrast, over an alphabet of size 3 or greater, f(x,y) can take any value between 0 and |x|+|y|, except 1; furthermore, we can always assume that the smaller string has only one type of letter.