New extremal binary self-dual codes of length 66 as extensions of self-dual codes over R_k

We obtain extremal binary self-dual codes of parameters [64,32,12] as binary images of self-dual codes over R₁, R₂ and R ₃ by employing different methods. We then apply the extension theorem to these codes to obtain a number of extremal binary self-dual codes of length 66 with trivial automorphism groups. Fifteen of the codes we obtain have new β values in W_66,3, of which only three were known to exist before. We also find nine codes with new β values in W_66,1, thus updating the list of such known codes.