射影一般线性群$\mathrm{PGL}_2(\mathrm{GF}(2^m))$与长度为$2^m+1$的线性码
The Projective General Linear Group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ and Linear Codes of Length $2^m+1$
摘要：
射影一般线性群$\mathrm{PGL}_2(\mathrm{GF}(2^m))$充当射影直线上的点集上的$3$ -传递置换群。本文的第一个目的是证明$\mathrm{GF}(2^h)$上所有在$\mathrm{PGL}_2(\mathrm{GF}(2^m))$下不变的线性码都是平凡码：重复码、整个空间$\mathrm{GF}(2^h)^{2^m+1}$及其对偶码。作为这个结果的一个应用，确定了在$\mathrm{PGL}_2(\mathrm{GF}(2^m))$下不变的所有$3$-$(q+1,k,λ)$设计的（0，1）-关联矩阵的$2$ -秩.第二个目标是给出$\mathrm{GF}(2^m)$上的两个无限循环码族，使得任意固定非零权的码字的支撑集在$\mathrm{PGL}_2(\mathrm{GF}(2^m))$下不变，因此任意非零权的码字支撑3-设计.来自第一系列的代码具有参数$[q+1,q-3,4]_q$，其中$q=2^m$和$m\ge 4$是偶数。确定了最小权值码字的确切数目，并且最小权值码字支持3- $(q+1,4,2)$设计。来自第二族的码具有参数$[q+1,4,q-4]_q$、$q=2^m$、甚至$m\ge 4$，并且最小权重码字支持3- $(q +1,q-4,(q-4)(q-5)(q-6)/60)$设计，其互补的3- $(q +1, 5, 1)$设计与具有这些参数的维特球面几何同构。证明了$\mathrm{GF}(q)$上线性码支持3- $(q +1,q-4,(q-4)(q-5)(q-6)/60)$设计的维数的一个下界，并证明了第二类码中最小权码字支持的设计满足这个下界.
英文摘要：
The projective general linear group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ acts as a $3$-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over $\mathrm{GF}(2^h)$ that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are trivial codes: the repetition code, the whole space $\mathrm{GF}(2^h)^{2^m+1}$, and their dual codes. As an application of this result, the $2$-ranks of the (0,1)-incidence matrices of all $3$-$(q+1,k,λ)$ designs that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are determined. The second objective is to present two infinite families of cyclic codes over $\mathrm{GF}(2^m)$ such that the set of the supports of all codewords of any fixed nonzero weight is invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$, therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters $[q+1,q-3,4]_q$, where $q=2^m$, and $m\ge 4$ is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3-$(q+1,4,2)$ design. A code from the second family has parameters $[q+1,4,q-4]_q$, $q=2^m$, $m\ge 4$ even, and the minimum weight codewords support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design, whose complementary 3-$(q +1, 5, 1)$ design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over $\mathrm{GF}(q)$ that can support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound.
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