Binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

07/22/2021

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by   Xia Li, et al.

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Goppa codes are particularly appealing for cryptographic applications. Every improvement of our knowledge of Goppa codes is of particular interest. In this paper, we present a sufficient and necessary condition for an irreducible monic polynomial g(x) of degree r over 𝔽_q satisfying γ g(x)=(x+d)^rg(A(x)), where q=2^n, A=([ a b; 1 d ])∈ PGL_2( F_q), ord(A) is a prime, g(a) 0, and 0γ∈ F_q. And we give a complete characterization of irreducible polynomials g(x) of degree 2s or 3s as above, where s is a positive integer. Moreover, we construct some binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes.