Characterization of $p$-ary functions in terms of association schemes and its applications

TL;DR

This paper establishes a criterion linking $p$-ary functions and association schemes via Walsh spectrum, characterizes when bent functions induce schemes, and constructs related codes and schemes with applications.

Contribution

It provides an explicit criterion for $p$-ary functions to generate association schemes and characterizes when bent functions induce schemes, advancing understanding of their structure and applications.

Findings

A criterion connecting Walsh spectrum and association schemes.

A proof that weakly regular $p$-ary bent functions induce $p$-class schemes.

Construction of infinite families of association schemes and two-weight codes.

Abstract

We obtain an explicit criterion for $p$-ary functions to produce association schemes in terms of their Walsh spectrum. Employing this characterization, we explicitly find a correlation between $p$-ary bent functions and association schemes; to be more exact, we prove that a $p$-ary bent function induces a $p$-class association scheme if and only if the function is weakly regular. As applications of our main criterion, we construct many infinite families of few-class association schemes arising from $p$-ary functions. Furthermore, we present four classes of $p$-ary two-weight linear codes, which are constructed from the association schemes produced in this paper.