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

Abstract

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.

Tang’s research was supported by The National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds). Qi was supported by Zhejiang provincial Natural Science Foundation of China (No. LY21A010013).