Association schemes arising from non-weakly regular bent functions

TL;DR

This paper constructs new infinite families of symmetric association schemes from non-weakly regular bent functions for any odd prime, expanding the known classes and providing conditions for their structure.

Contribution

It introduces novel constructions of symmetric association schemes with classes related to any odd prime p using non-weakly regular bent functions, extending previous work.

Findings

Constructed infinite families of association schemes with classes 2p, 2p+1, and (3p+1)/2 for odd prime p.

Fused association schemes yield t-class schemes with t=4,5,6,7.

Provided necessary and sufficient conditions for partitions to induce symmetric association schemes.

Abstract

Association schemes play an important role in algebraic combinatorics and have important applications in coding theory, graph theory and design theory. The methods to construct association schemes by using bent functions have been extensively studied. Recently, in [13], {\"O}zbudak and Pelen constructed infinite families of symmetric association schemes of classes $5$ and $6$ by using ternary non-weakly regular bent functions.They also stated that constructing $2p$-class association schemes from $p$-ary non-weakly regular bent functions is an interesting problem, where $p>3$ is an odd prime. In this paper, using non-weakly regular bent functions, we construct infinite families of symmetric association schemes of classes $2p$, $(2p+1)$ and $\frac{3p+1}{2}$ for any odd prime $p$. Fusing those association schemes, we also obtain $t$-class symmetric association schemes, where $t=4,5,6,7$. In addition, we give the sufficient and necessary conditions for the partitions $P$, $D$, $T$, $U$ and $V$ (defined in this paper) to induce symmetric association schemes.