De Bruijn Sequences with Minimum Discrepancy

TL;DR

This paper determines the minimal discrepancy achievable by binary de Bruijn sequences of order n, proving it to be n, and provides an algorithm to construct such sequences, extending to arbitrary alphabets with near-optimal discrepancy.

Contribution

It solves the open problem of minimal discrepancy for binary de Bruijn sequences and introduces an algorithm for constructing sequences with this minimal discrepancy.

Findings

Minimal discrepancy for binary de Bruijn sequences is n.

An algorithm constructs sequences achieving this minimal discrepancy.

Extension to arbitrary alphabets yields sequences with discrepancy at most n+1.

Abstract

The discrepancy of a binary string is the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. In this note we determine the minimal discrepancy that a binary de Bruijn sequence of order $n$ can achieve, which is $n$. This was an open problem until now. We give an algorithm that constructs a binary de Bruijn sequence with minimal discrepancy. A slight modification of this algorithm deals with arbitrary alphabets and yields de Bruijn sequences of order $n$ with discrepancy at most $1$ above the trivial lower bound $n$.