Association schemes arising from non-weakly regular bent functions

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 Özbudak and Pelen (J Algebr Comb 56:635–658, 2022), Özbudak 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 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.