| In the field of algebraic geometry, algebraic curves play an important role in coding theory. In this work we will work in finite fields Fq2 where q = pn for some odd prime p. We will work with hyperelliptic curves denoted YS: y2 = fS(x) = s∈S (x - s) where S ⊂ Fq2 ; XS represents the non-singular projective model of YS. We utilize the Hasse-Weil bound | XS( Fq2 ) - (q2 + 1)| ≤ 2 gq and Λ(S) = b∈Fq2 chi(fs(b)) = |QS | - |Q'S| as we try to determine when XS is maximal, minimal and/or optimal.;We say XS is maximal if the upper Hasse-Weil bound is achieved and if ∣XS&parl0;Fq2 &parr0;∣= 1+q2+qS -1if Sis odd,1+q2+q S-2 ifS iseven. We say XS is minimal if the lower Hasse-Weil bound is reached and if ∣XS&parl0;Fq2 &parr0;∣= 1+q2-qS -1if Sis odd,1+q2-q S-2 ifS is even. Finally we say XS is optimal if Λ(S) = q2 - |S|. There is an overlap of maximal curves and minimal curves; there is also an overlap between maximal curves and optimal curves. In particular maximal sets, S, and their associated maximal curves XS are useful in coding theory. |
