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

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.