2022/01/11 by Isabel Byrne, Byrne, Isabel, Natalie Dodson +7 · 1 citation
Computer Science · Biochemistry, Genetics and Molecular Biology · #Coding theory and cryptography #Cellular Automata and Applications #DNA and Biological Computing
paper · pdf · doi:10.48550/arxiv.2201.03741
In this article we prove that a class of Goppa codes whose Goppa polynomial is of the form g(x) = x + xq + ⋯ + x^qm-1 where m ≥ 3 (i.e. g(x) is a trace polynomial from a field extension of degree m ≥ 3) has a better minimum distance than what the Goppa bound d ≥ 2deg(g(x))+1 implies. Our improvement is based on finding another Goppa polynomial h such that C(L,g) = C(M, h) but deg(h) > deg(g). This is a significant improvement over Trace Goppa codes over quadratic field extensions (i.e. the case m = 2), as the Goppa bound for the quadratic case is sharp.