A note on diagonal and Hermitian surfaces

TL;DR

This paper reviews properties, enumeration, and construction methods for points on diagonal and Hermitian surfaces, highlighting their relevance to coding theory and providing insights into their zeta functions and recursive point constructions.

Contribution

It offers a concise overview of existing results and discusses recursive techniques for constructing rational points on Hermitian surfaces, connecting these to code construction.

Findings

Zeta function of diagonal surfaces derived from Wolfmann's work

Recursive methods for Hermitian surface points explored

Implications for surface-based code construction noted

Abstract

Aspects of the properties, enumeration and construction of points on diagonal and Hermitian surfaces have been considered extensively in the literature and are further considered here. The zeta function of diagonal surfaces is given as a direct result of the work of Wolfmann. Recursive construction techniques for the set of rational points of Hermitian surfaces are of interest. The relationship of these techniques here to the construction of codes on surfaces is briefly noted.