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).
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Zhao, Y., Tang, C., Qi, Y. (2023). The Projective General Linear Group PGL(2, \(5^m\)) and Linear Codes of Length \(5^m+1\). In: Mesnager, S., Zhou, Z. (eds) Arithmetic of Finite Fields. WAIFI 2022. Lecture Notes in Computer Science, vol 13638. Springer, Cham. https://doi.org/10.1007/978-3-031-22944-2_11
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