The Projective General Linear Group PGL(2, $$5^m$$ ) and Linear Codes of Length $$5^m+1$$

The importance of interactions between groups, linear codes and t-designs has been well recognized for decades. Linear codes that are invariant under groups acting on the set of code coordinates have found important applications for the construction of combinatorial t-designs. Examples of such codes are the Golay codes, the quadratic-residue codes, and the affine-invariant codes. Let $$q=5^m$$ . The projective general linear group PGL(2, q) acts as a 3-transitive permutation group on the set of points of the projective line. This paper is to present two infinite families of cyclic codes over GF $$(5^m)$$ such that the set of the supports of all codewords of any fixed nonzero weight is invariant under PGL(2, q), therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters $$[q+1,4,q-5]_q$$ , where $$q=5^m$$ , and $$ m\ge 2$$ . A code from the second family has parameters $$[q+1,q-3,4]_q$$ ,  $$q=5^m,~m\ge 2$$ . This paper also points out that the set of the support of all codewords of these two kinds of codes with any nonzero weight is invariant under $$\textrm{Stab}_{U_{q+1}}$$ , thus the corresponding incidence structure supports 3-design.