The Projective General Linear Group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ and Linear Codes of Length $2^m+1$

Abstract

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.