Let \(q=2^m\). The projective general linear group \({\mathrm {PGL}}(2,q)\) 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 \({\mathrm {PGL}}(2,q)\) 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 \(3-(q+1,k,\lambda )\) designs that are invariant under \({\mathrm {PGL}}(2,q)\) 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 \({\mathrm {PGL}}(2,q)\), 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.

令\（q = 2 ^ m \）。投影一般线性群\（{\ mathrm {PGL}}（2，q）\）在投影线的点集上充当3个传递矩阵。本文的第一个目标是证明\（{\ mathrm {GF}}（2 ^ h）\）上的所有线性代码在\（{\ mathrm {PGL}}（2，q）\）下不变是平凡的代码：重复代码，整个空间\（{\ mathrm {GF}}（2 ^ h）^ {2 ^ m + 1} \）及其对偶代码。作为该结果的应用，所有\（3-（q + 1，k，\ lambda）\）设计在（（{\ mathrm { PGL}}（2，q）\）确定。第二个目标是在\（{\ mathrm {GF}}（2 ^ m）\）上给出两个无限的循环码族，使得任何固定非零权重的所有码字的支持集在\（{ \ mathrm {PGL}}（2，q）\），因此，任何非零权重的代码字都支持3设计。第一个族的代码具有参数\（[q + 1，q-3,4] _q \），其中\（q = 2 ^ m \）和\（m \ ge 4 \）是偶数。确定最小权重的码字的确切数目，并且最小权重的码字支持3- \（（q + 1,4,2）\）设计。第二族的代码具有参数\（[q + 1,4，q-4] _q \），\（q = 2 ^ m \），\（m \ ge 4 \）甚至最小权重码字支持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Witt球形几何同构。的下限的线性码的尺寸超过\（{\ mathrm {GF}}（Q）\） ，可支持3- \（（Q + 1，Q-4，（Q-4）（Q-证明了5）（q-6）/ 60）\）设计，并且表明第二个代码系列的代码中最小权重的代码字支持的设计满足此限制。