Codes defined from some maximal curves (Q804543)

For every prime power q, the authors construct a family of Goppa codes over \({\mathbb{F}}_{q^ 2}\) by means of the plane curve C given by \(X^{q+1}=Y^ qZ+YZ^ q\). This smooth curve has genus (q-1)q/2 and it has the largest number of \({\mathbb{F}}_{q^ 2}\)-rational points, viz. \(q^ 3+1\) a curve of genus (q-1)q/2 can have. The codes have length \(q^ 3\) and are the images of the spaces \(L(m(Q))=\{f\in {\mathbb{F}}_{q^ 2}(C):\;(f)>-m(Q)\}\) for \(q^ 2-q-2<m<q^ 3.\) Here Q denotes the point (0:1:0) of C. The parameters of the codes are estimated. The automorphisms of the curve that fix Q, give rise to a group of automorphisms of the code. The structure of this group is determined. Finally the authors remark that their results are similar to \textit{H. J. Tiersma}'s results [IEEE Trans. Inf. Theory IT-33, 605-609 (1987; Zbl 0627.94019)] on codes constructed by means of the curve \(X^{q+1}+Y^{q+1}=Z^{q+1}\), since their curve C is isomorphic to this Fermat curve.