Some codes and designs invariant under the groups $S_7$ and $S_8$

Document Type : Research Paper

Author

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

Abstract

We use the Key-Moori Method 1 and examine 1-designs and codes from the representations of the alternating group $A_7$. It is shown that a self-dual symmetric 2-$(35,18,9)$ design and an optimal even binary $[21,14,4]$ LCD code are found such that they are invariant under the full automorphism groups $S_8$ and $S_7$, respectively. Moreover, designs with parameters 1-$(21,l,k_{1,l})$ and 1-$(35,l,k_{2,l})$ are obtained, where $\omega$ is a codeword, $l=wt(\omega)$, $k_{1,l}=l|\omega^{S_7}|/21$ and $k_{2,l}=l|\omega^{S_7}|/35$. It is seen that there exist a 2-$(21,5,12)$ design with the full automorphism group $S_7$ among these 1-designs.

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References

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Journal of Mahani Mathematical Research
Volume 13, Issue 1 - Serial Number 26
November 2023
Pages 511-524

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History

  • Receive Date: 08 April 2023
  • Revise Date: 09 September 2023
  • Accept Date: 15 October 2023

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