# Quaternary codes and a class of 2-designs invariant under the group $A_8$

Document Type : Research Paper

Author

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

10.29252/as.2021.2254

Abstract

In this paper, we use the Key-Moori Method 1 and construct a quaternary code $\mathcal{C}_8$ from a primitive representation of the group $PSL_2(9)$ of degree 15. We see that $\mathcal{C}_8$ is a self-orthogonal even code with the automorphism group isomorphic to the alternating group $A_8$. It is shown that by taking the support of any codeword $\omega$ of weight $l$ in $\mathcal{C}_8$ or $\mathcal{C}_8^\bot$, and orbiting it under $A_8$, a 2-$(15,l,\lambda)$ design invariant under the group $A_8$ is obtained, where $\lambda=\binom{l}{2}|\omega^{A_8}|/\binom{15}{2}$. A number of these designs have not been known before up to our best knowledge. The structure of the stabilizers $(A_8)_\omega$ is determined and moreover, primitivity of $A_8$ on each design is examined.

Keywords

#### References

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