In this paper we prove that the class of Goppa codes whose Goppa polynomial is of the form \(g(x) = \textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}\) where \(\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}\) is a trace polynomial from a field extension of degree \(m \ge 3\) has a better minimum distance than what the Goppa bound \(d \ge 2\deg (g(x))+1\) implies. This result is a significant improvement compared to the minimum distance of Trace Goppa codes over quadratic field extensions (the case \(m = 2\)). We present two different techniques to improve the minimum distance bound. For general p we prove that the Goppa code \(C(L, \textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}})\) is equivalent to another Goppa code C(M, h) where \(\deg (h) > \deg (\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}})\). For \(p=2\) we use the fact that the values of \(\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}\) are fixed under q–powers to find several new parity check equations which increase the known distance bounds.

在本文中，我们证明了 Goppa 码的类别，其 Goppa 多项式的形式为\(g(x) = \textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}\) 其中 \(\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q }}\) 是来自 \(m \ge 3\) 度场扩展的迹多项式，其最小距离比 Goppa 边界 \(d \ge 2\deg (g(x))+1 \ )所暗示的更好。与二次域扩展上的 Trace Goppa 码的最小距离相比（ \(m = 2\)情况），这个结果是一个显着的改进。我们提出了两种不同的技术来改进最小距离界限。对于一般的p，我们证明 Goppa 代码\(C(L, \textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}})\) 等价于另一个 Goppa 代码 C ( M , h )，其中 \(\deg (h) > \deg (\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}} _ { q } }  ) \ )。对于\(p=2\)，我们利用 \(\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}\) 的值在 q 幂下固定的事实来找到几个新的奇偶校验方程，这些方程增加了已知的距离范围。