## Suzuki Function Field

Let r be a positive integer and let q = 2r and b = 2q2. The Suzuki function field over Fb  is an algebraic function field defined by F = Fb(x, y) with

yb + y = xq(xb + x).

It corresponds to the projective plane curve defined by the equation

Zq(Yb + YZb−1) = Xq(Xb + XZb−1).

It has genus g(F/Fb) = q(b−1) and N(F/Fb) = b2 + 1 rational points, namely P := (0 : 1 : 0) and Px,y := (x : y : 1) for all x, yFb.

### Optimality

The Suzuki function field is optimal because the number of rational places meets Serre’s improved Hasse-Weil bound.

### The Weierstrass Semigroup

The Weierstrass semigroup of P is generated by {b, b + q, b + 2q, b + 2q + 1}. Thus the g = 14 gap numbers for b = 8 are {1, 2, 3, 4, 5, 6, 7, 9, 11, 14, 15, 17, 19, 27}. A basis of the Riemann-Roch space L(rP) can also be found in .

### Improvements for Suzuki Codes

It is shown in  that the true minimum distance of the algebraic-geometric codes derived from the Suzuki curve is d = 56 for r = 10 (i.e., a [64, 3, 56]-code) and d = 42 for r = 23 (i.e., a [64, 11, 42]-code).