On a Connection between Ideal Two-level Autocorrelation and Almost Balancedness of $p$-ary Sequences

TL;DR

This paper establishes a link between ideal two-level autocorrelation in p-ary sequences and the almost balanced distribution of elements, revealing the existence of a uniquely less frequent element in the sequence.

Contribution

It proves the existence of a special element in p-ary sequences with ideal autocorrelation, extending understanding of their distribution properties.

Findings

Existence of an element appearing once less than others in the sequence.

Such an element can be zero or any element of the field.

The result is proved using algebraic methods.

Abstract

In this correspondence, for every periodic $p-$ary sequence satisfying ideal two-level autocorrelation property the existence of an element of the field ${\bf GF}(p)$ which appears one time less than all the rest that are equally distributed in a period of that sequence, is proved by algebraic method. In addition, it is shown that such a special element might not be only the zero element but as well arbitrary element of that field.