The projective general linear group PGL (2 , 2 m) and linear codes of length 2 m+ 1

Document Type

Article

Publication Date

5-21-2021

Department

Department of Mathematical Sciences

Abstract

Let q= 2 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. The first objective of this paper is to prove that all linear codes over GF (2 h) that are invariant under PGL (2 , q) are trivial codes: the repetition code, the whole space GF(2h)2m+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 , q) 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 , 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, 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.

Publication Title

Designs, Codes, and Cryptography

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