Abstract
In this paper we consider the Hermitian curve \(y^{q_0} + y = x^{q_0+1}\) over the field \({\mathbb {F}}_q (q:=q_0^2)\). The automorphism group of this curve is known to be the projective unitary group \(\text {PGU}(3,q)=\mathrm{{ ^2A_2}}(q)\) with \(q_0^3(q_0^3+1)(q_0^2-1)\) elements. We follow the construction done for the Suzuki code in Eid et al. (Designs Codes Cryptogr 81(3):413–425, 2016, https://doi.org/10.1007/s10623-015-0164-5). We construct algebraic geometry codes over \({\mathbb {F}}_{q^3}\) from an \(\mathrm{{ ^2A_2}}(q)\)-invariant divisor D, give an explicit basis for the Riemann–Roch space \(L(\ell D)\) for \(0 < \ell \le q_0^3-1\). These families of codes have good parameters and information rate close to one. In addition, they are explicitly constructed. The dual codes of these families are of the same kind if \(q_0^3-2g+1 \le \ell \le q_0^3-1\).
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Communicated by Sergio R. López-Permouth.
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Eid, A. Hermitian-invariant codes from the Hermitian curve. São Paulo J. Math. Sci. 17, 927–939 (2023). https://doi.org/10.1007/s40863-022-00315-x
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DOI: https://doi.org/10.1007/s40863-022-00315-x