论文标题
投射通用线性组$ \ mathrm {pgl} _2（\ mathrm {gf}（2^m））$和长度的线性代码$ 2^m+1 $
The Projective General Linear Group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ and Linear Codes of Length $2^m+1$
论文作者
论文摘要
投影通用线性组$ \ mathrm {pgl} _2（\ mathrm {gf}（2^m））$在投影线的一组点上充当$ 3 $转换的置换组。本文的第一个目的是证明$ \ mathrm {gf}（2^h）$上的所有线性代码在$ \ mathrm {pgl} _2（\ mathrm {gf}（gf}（2^m）下$ \ mathrm {gf}（2^h）^{2^m+1} $及其双重代码。作为此结果的应用，所有$ 3 $ - $（q+1，k，λ）$ 2 $ lanks的含量矩阵确定了不变的$ \ mathrm {pgl} _2 _2（\ mathrm {gf}（gf}（2^m））$。第二个目标是在$ \ mathrm {gf}（2^m）$上呈现两个无限的环状代码家族，以使任何固定非零重量的所有代码字的支撑集在$ \ mathrm {pgl}下是不变的，第一个家庭的代码具有参数$ [q+1，q-3,4] _q $，其中$ q = 2^m $，$ m \ ge 4 $均匀。确定了最小重量的代码字的确切数量，并且最小重量的代码字支持3- $（Q+1,4,2）$设计。 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这些参数的球形几何形状。在$ \ mathrm {gf}（q）$上的线性代码的尺寸上，可以支持3- $（Q +1，Q-4，（Q-4）（Q-4）（Q-6）（Q-6）/60）$设计，并且证明了该设计由代码范围最低的代码范围的代码范围所支撑的设计，这表明了第二个代码范围的代码范围。
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.
