Autocorrelation-Run Formula for Binary Sequences

TL;DR

This paper proves that for binary sequences, the autocorrelation function is entirely determined by the sequence's run structure, linking two fundamental concepts used in randomness testing.

Contribution

It establishes a complete theoretical connection between autocorrelation and run structure in binary sequences, which were previously considered independent.

Findings

Autocorrelation function is determined by run structure.

Run structure fully characterizes autocorrelation in binary sequences.

Provides theoretical insight into binary sequence analysis.

Abstract

The autocorrelation function and the run structure are two basic notions for binary sequences, and have been used as two independent postulates to test randomness of binary sequences ever since Golomb 1955. In this paper, we prove for binary sequence that the autocorrelation function is in fact completely determined by its run structure.