# Extending de Bruijn sequences to larger alphabets

A circular de Bruijn sequence of order n in an alphabet of k symbols is a sequence in which each sequence of length n occurs exactly once. In this work we show that for each circular de Bruijn sequence v of order n in an alphabet of k symbols there is another circular de Bruijn sequence w also of order n in an alphabet with one more symbol, that is an alphabet of k + 1 symbols, such that v is a subsequence of w and in between any two successive occurrences of the new symbol in w there are at most n + 2k-2 consecutive symbols of v. We give an algorithm that receives as input such a sequence v and outputs a sequence w. We also give a much faster algorithm that receives as input such a sequence v and outputs a sequence w, but the new symbol may not be evenly spread out.

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