论文标题
扩展不可约的二进制GOPPA代码的数量
The number of extended irreducible binary Goppa codes
论文作者
论文摘要
GOPPA在1970年代发现了代数几何形状与代码之间的关系，这导致了GOPPA代码家族。作为线性代码最有趣的子类之一，GOPPA代码家族通常被选为McEliece Cryptosystem中的关键。了解固定参数的不相等二进制GOPPA代码的数量可能有助于评估这种加密系统的安全性。令$ n \ geq5 $为一个奇数的数字，让$ q = 2^n $，让$ r \ geq3 $为一个正整数满足$ \ gcd（r，n）= 1 $。本文的目的是建立上层限制的不等性延长的不可约定的二进制GOPPA代码$ q+1 $ and $ r $。用于此目的的潜在数学对象是计算投射半线性组的轨道数量gal}（\ Mathbb {f} _ {q^r}/\ Mathbb {f} _2）$上的所有nonic norig norredibible $ r $ $ r $的$ \ mathcal {i} _r $ the有限字段$ \ mathbb {f} _q $ $ r $的所有monicribible $ r $。 $ {\ rm pgl} _2（\ Mathbb {f} _Q）\ rtimes {\ rm gal}（\ Mathbb {f} _ {q^r}/\ m mathbb {f} $ n $ n $ n $ n $ n $ rtimes n $ rtimes {上限为不相等的不可约定的二进制GOPPA代码的长度$ q+1 $和度$ r $的上限。我们的主要结果自然包含Ryan（IEEE-TIT 2015），Huang and Yue（IEEE-TIT，2022年）和Chen和Zhang（IEEE-TIT，2022）的主要结果，这些案例$ r = 4 $，$ r = 4 $，$ r = 6 $和$ \ gcd（r，q^3-q）（r，q^3-q）= 1 $ $ $。
Goppa, in the 1970s, discovered the relation between algebraic geometry and codes, which led to the family of Goppa codes. As one of the most interesting subclasses of linear codes, the family of Goppa codes is often chosen as a key in the McEliece cryptosystem. Knowledge of the number of inequivalent binary Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. Let $n\geq5$ be an odd prime number, let $q=2^n$ and let $r\geq3$ be a positive integer satisfying $\gcd(r,n)=1$. The purpose of this paper is to establish an upper bound on the number of inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree $r$.A potential mathematical object for this purpose is to count the number of orbits of the projective semi-linear group ${\rm PGL}_2(\mathbb{F}_q)\rtimes{\rm Gal}(\mathbb{F}_{q^r}/\mathbb{F}_2)$ on the set $\mathcal{I}_r$ of all monic irreducible polynomials of degree $r$ over the finite field $\mathbb{F}_q$. An explicit formula for the number of orbits of ${\rm PGL}_2(\mathbb{F}_q)\rtimes{\rm Gal}(\mathbb{F}_{q^r}/\mathbb{F}_2)$ on $\mathcal{I}_r$ is given, and consequently, an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree $r$ is derived. Our main result naturally contains the main results of Ryan (IEEE-TIT 2015), Huang and Yue (IEEE-TIT, 2022) and, Chen and Zhang (IEEE-TIT, 2022), which considered the cases $r=4$, $r=6$ and $\gcd(r,q^3-q)=1$ respectively.
