Some $3$-designs invariant under $2.P{\Sigma}L(2,49).$

TL;DR

This paper constructs a specific ternary code from a Jacobsthal matrix, explores its automorphism group related to a projective special linear group, and derives new 3-designs from these codes and their extensions.

Contribution

It introduces a new class of GQR codes invariant under a group related to $P{ ext{Sigma}}L(2,49)$ and links these codes to the construction of 3-designs with automorphism groups.

Findings

Constructed a [49,25,7] GQR code from Jacobsthal matrix.

Derived a 3-(50,14,1248) design from the code and its extension.

Identified the automorphism group as a double cover of $P{ ext{Sigma}}L(2,49)$.

Abstract

We construct a ternary [49,25,7] code from the row span of a Jacobsthal matrix. It is equivalent to a Generalized Quadratic Residue (GQR) code in the sense of van Lint and MacWilliams (1978). These codes are the abelian generalizations of the quadratic residue (QR) codes which are cyclic. The union of the [50,25,8] extension of the said code and its dual supports a 3-(50,14,1248) design. The automorphism group of the latter design is a double cover of the permutation part of the automorphism group of the [50,25,8] code, which is isomorphic to $P{\Sigma}L(2,49).$ Other weights in this code, other GQR codes, and other QR codes yield other 3-designs by the same process. A simple group action argument is provided to explain this behaviour of isodual codes.