Suzuki-invariant codes from the Suzuki curve

In this paper we consider the Suzuki curve \(y^q + y = x^{q_0}(x^q + x)\) over the field with \(q = 2^{2m+1}\) elements. The automorphism group of this curve is known to be the Suzuki group \(\mathrm{{Sz}}(q)\) with \(q^2(q-1)(q^2+1)\) elements. We construct AG codes over \(\mathbb {F}_{q^4}\) from an \(\mathrm{{Sz}}(q)\)-invariant divisor D, giving an explicit basis for the Riemann–Roch space \(L(\ell D)\) for \(0 < \ell \le q^2-1\). The full Suzuki group \(\mathrm{{Sz}}(q)\) acts faithfully on each code. These families of codes have very good parameters and information rate close to 1. In addition, they are explicitly constructed. The dual codes of these families are of the same kind if \(2g-1 \le \ell \le q^2-1\).