Bounds for the discrete correlation of infinite sequences on k symbols and generalized Rudin-Shapiro sequences

TL;DR

This paper investigates the discrete correlation properties of infinite sequences over finite alphabets, establishing bounds on their similarity and providing explicit constructions that maximize differences, inspired by Rudin-Shapiro sequences.

Contribution

It introduces bounds for the discrete correlation of sequences and constructs sequences that achieve these bounds, extending Rudin-Shapiro sequences to a broader context.

Findings

Sequences cannot be too dissimilar based on combinatorial bounds.

Explicit constructions achieve maximum possible difference in correlation.

Results generalize properties of Rudin-Shapiro sequences.

Abstract

Motivated by the known autocorrelation properties of the Rudin-Shapiro sequence, we study the discrete correlation among infinite sequences over a finite alphabet, where we just take into account whether two symbols are identical. We show by combinatorial means that sequences cannot be "too" different, and by an explicit construction generalizing the Rudin-Shapiro sequence, we show that we can achieve the maximum possible difference.