The projective general linear group \n \n \n \n $${\\mathrm {PGL}}(2,2^m)$$\n \n \n PGL\n (\n 2\n ,\n \n 2\n

Abstract

Let $$q=2^m$$ q = 2 m . The projective general linear group $${\\mathrm {PGL}}(2,q)$$ 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 $${\\mathrm {GF}}(2^h)$$ GF ( 2 h ) that are invariant under $${\\mathrm {PGL}}(2,q)$$ PGL ( 2 , q ) are trivial codes: the repetition code, the whole space $${\\mathrm {GF}}(2^h)^{2^m+1}$$ 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 $$3-(q+1,k,\\lambda )$$ 3 - ( q + 1 , k , λ ) designs that are invariant under $${\\mathrm {PGL}}(2,q)$$ PGL ( 2 , q ) are determined. The second objective is to present two infinite families of cyclic codes over $${\\mathrm {GF}}(2^m)$$ GF ( 2 m ) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under $${\\mathrm {PGL}}(2,q)$$ PGL ( 2 , q ) , therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters $$[q+1,q-3,4]_q$$ [ q + 1 , q - 3 , 4 ] q , where $$q=2^m$$ q = 2 m , and $$m\\ge 4$$ m ≥ 4 is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3- $$(q+1,4,2)$$ ( q + 1 , 4 , 2 ) design. A code from the second family has parameters $$[q+1,4,q-4]_q$$ [ q + 1 , 4 , q - 4 ] q , $$q=2^m$$ q = 2 m , $$m\\ge 4$$ m ≥ 4 even, and the minimum weight codewords support a 3- $$(q +1,q-4,(q-4)(q-5)(q-6)/60)$$ ( q + 1 , q - 4 , ( q - 4 ) ( q - 5 ) ( q - 6 ) / 60 ) design, whose complementary 3- $$(q +1, 5, 1)$$ ( 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 $${\\mathrm {GF}}(q)$$ GF ( q ) that can support a 3- $$(q +1,q-4,(q-4)(q-5)(q-6)/60)$$ ( 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.