The Projective General Linear Group PGL_2(GF(2^m)) and Linear Codes of Length 2^m+1
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The projective general linear group PGL_2(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 GF(2^h) that are invariant under PGL_2(GF(2^m)) are trivial codes: the repetition code, the whole space 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 PGL_2(GF(2^m)) are determined. The second objective is to present two infinite families of cyclic codes over GF(2^m) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under PGL_2(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≥ 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≥ 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 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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