LocalNR package and some of its applications

Abstract

In 2024, the authors developed well-known algorithms for the construction and analysis of finite nearrings, implementing it to compute local nearrings of order greater than 31. The new package was named LocalNR package. In this paper, using our method we

• indicate seventeen \(2\)-generated groups of exponent 8 which are the additive groups of zero-symmetric local nearrings of order 128; and formulate a conjecture on all such nearrings according to which there left only 2 open cases;
• prove that there are no local nearrings with the Miller-Moreno multiplicative group of order 64 and the additive group to be a \(2\)-generated group of order 128.