A quotient curve of the Fermat curve of degree twenty-three attaining the Serre bound

Abstract

The curve in the title is a non-maximal curve of genus 11, which has a defining equation analogous to the Klein quartic's. We study its properties and show how to apply it to coding theory.In 1970s, Goppa discovered algebraic geometric codes in [9]. His theory gives us a guarantee on the existence of efficient codes, when we have curves with many rational points for a fixed genus and a fixed finite field. It creates strong interest on such curves; see [8,18]. By a curve we mean a smooth absolutely irreducible projective curve.For the number of rational points on a curve C of genus g over a finite field F q , the Hasse-Weil boundis well-known. A curve is said to be maximal over F q , if it attains this bound. After the appearance of algebraic geometric codes, in 1983 Serre sharpened this bound in [20],