Generator polynomials of cyclic expurgated or extended Goppa codes
Xue Jia, Fengwei Li, Huan Sun, Qin Yue
TL;DR
The paper addresses the problem of determining generator polynomials for cyclic expurgated and extended Goppa codes under prescribed permutations from $A\in PGL_2(\mathbb F_q)$, framing cyclic codes as ideals in $\mathbb F_q[x]/(x^n-1)$ and situating Goppa codes within code-based cryptography. The authors leverage orbit partitions of the projective line under a cyclic subgroup of $PGL_2(\mathbb F_q)$ (and related $P\Gamma L_2$ actions) to derive explicit generator polynomials for various Goppa polynomials $g(x)$, with a focus on the case $\sigma^j=1$. Key results include explicit generators for $g(x)=g_1(x)$ or $g_2(x)$, and for powers $g_1(x)^s$ or $g_2(x)^t$, showing that these codes are BCH codes with designed distances $\delta=s+2$ or $\delta=t+2$, respectively, and providing the general product case $g_1(x)^s g_2(x)^t$ via a least-common-multiple of the individual generators. The work yields concrete, implementable generator polynomials such as $u_1(x)=(x+1)m_{\rho^{-1}}(x)$ and $u_2(x)=(x+1)m_{\rho}(x)$, and extends to irreducible settings over extension fields when necessary. Overall, the paper delivers a complete generator-polynomial characterization for the $\sigma^j=1$ regime, enabling explicit construction of cyclic expurgated/extended Goppa codes with cyclic extensions and informing potential cryptographic applications, while leaving the $\sigma^j\neq 1$ case as future work.
Abstract
Classical Goppa codes are a well-known class of codes with applications in code-based cryptography, which are a special case of alternant codes. Many papers are devoted to the search for Goppa codes with a cyclic extension or with a cyclic parity-check subcode. Let $\Bbb F_q$ be a finite field with $q=2^l$ elements, where $l$ is a positive integer. In this paper, we determine all the generator polynomials of cyclic expurgated or extended Goppa codes under some prescribed permutations induced by the projective general linear automorphism $A \in PGL_2(\Bbb F_q)$. Moreover, we provide some examples to support our findings.