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

2021/07/22 by Li, Xia, Yue, Qin, Huang, Daitao
#FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2107.10494

Abstract

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 \mathbbF_q satisfying γg(x)=(x+d)^rg(A(x)), where q=2ⁿ, A=(a · amp;b
1 · amp;d)∈ PGL₂(\Bbb F_q), ord(A) is a prime, g(a)≠ 0, and 0≠ γ∈ \Bbb 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.

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