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Published in:
23-04-2023
Improving the minimum distance bound of Trace Goppa codes
Published in: Designs, Codes and Cryptography | Issue 8/2023
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Abstract
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.