Goppa codes on a family of algebraic number fields

M. M. Glukhov (jr.)

Diskretnaya Matematika, Tome 13 (2001) no. 2, pp. 14-34

Résumé

We describe some properties of the geometric Goppa codes on the curve determined by the equation $$ y^s=(x^{q^{(n-u)/2}-1}+1)^a (x^{q^{(n+u)/2}-1}+1)^b $$ over a finite field $K=F_{q^n}$ with an arbitrary odd $q$, $n>1$, where $s=a+b$, $s\mid q-1$, $u=1$ for odd $n$ and $u=2$ for even $n$. We find the number of the $F_{q^n}$-rational points of the curve and the degrees and ramification indexes of the maximal ideals of the discrete valuation rings of the field $K(x,y)$. In some cases, the bases of the codes are found.