Geodesic
Goppa codes on a family of algebraic number fields
Diskretnaya Matematika, Tome 13 (2001) no. 2, pp. 14-34 
M. M. Glukhov (jr.). Goppa codes on a family of algebraic number fields. Diskretnaya Matematika, Tome 13 (2001) no. 2, pp. 14-34. http://geodesic.mathdoc.fr/item/DM_2001_13_2_a1/
@article{DM_2001_13_2_a1,
     author = {M. M. Glukhov (jr.)},
     title = {Goppa codes on a family of algebraic number fields},
     journal = {Diskretnaya Matematika},
     pages = {14--34},
     year = {2001},
     volume = {13},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2001_13_2_a1/}
}
TY  - JOUR
AU  - M. M. Glukhov (jr.)
TI  - Goppa codes on a family of algebraic number fields
JO  - Diskretnaya Matematika
PY  - 2001
SP  - 14
EP  - 34
VL  - 13
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/DM_2001_13_2_a1/
LA  - ru
ID  - DM_2001_13_2_a1
ER  - 
%0 Journal Article
%A M. M. Glukhov (jr.)
%T Goppa codes on a family of algebraic number fields
%J Diskretnaya Matematika
%D 2001
%P 14-34
%V 13
%N 2
%U http://geodesic.mathdoc.fr/item/DM_2001_13_2_a1/
%G ru
%F DM_2001_13_2_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Glukhov M. M. (ml.), “Nizhnie otsenki summ kharakterov ot mnogochlenov nad konechnymi polyami”, Diskretnaya matematika, 6:3 (1994), 136–142 | MR | Zbl

[2] Glukhov M. M. (ml.), “O kanonicheskikh razlozheniyakh nekotorykh dvuchlenov nad $GF(q^m)$”, Tezisy dokl. II mezhdunarodnoi konferentsii “Algebraicheskie, veroyatnostnye, geometricheskie, kombinatornye i funktsionalnye metody v teorii chisel”, Voronezh, 1995, 41

[3] Lidl R., Niderraiter G., Konechnye polya, Mir, Moskva, 1988 | Zbl

[4] Ozbudak F., Glukhov M., Jr., “Codes on superelliptic curves”, Turkish J. Math., 22 (1998), 223–234 | MR

[5] Stepanov S. A., Arifmetika algebraicheskikh krivykh, Nauka, Moskva, 1991 | MR

[6] Stepanov S. A., Ozbudak F., “Rassloennye proizvedeniya giperellipticheskikh krivykh i geometricheskie kody Goppy”, Diskretnaya matematika, 9:3 (1997), 36–42 | MR | Zbl

[7] Stichtenoth H., Algebraic Function Fields and Codes, Springer, Berlin, 1993 | MR | Zbl