SOME CODES AND DESIGNS INVARIANT UNDER THE GROUPS S7 AND S8

We use the Key-Moori Method 1 and examine 1-designs and codes from the representations of the alternating group A7. It is shown that a self-dual symmetric 2-(35, 18, 9) design and an optimal even bi-nary [21, 14, 4] LCD code are found such that they are invariant under the full automorphism groups S8 and S7, respectively. Moreover, designs with parameters 1-(21, l, k1,l) and 1-(35, l, k2,l) are obtained, where ω is a codeword, l = wt(ω), k1,l = l|ωS7 |/21 and k2,l = l|ωS7 |/35. It is seen that there exist a 2-(21, 5, 12) design with the full automorphism group S7 among these 1-designs.