New extremal binary self-dual codes of length 68

In this correspondence, we consider quadratic double and bordered double circulant construction methods over the ring R := F2 + uF2 + u2F2, where u3 = 1. Among other examples, extremal binary self-dual codes of length 66 are obtained by these constructions. These are extended by using extension theorems for self-dual codes and as a result 8 new extremal binary self-dual codes of length 68 are obtained. More precisely, codes with β=117, 120, 133 in W68;1 and with γ = 1, β=49, 57, 59 and codes with γ=2, β=69, 81 in W68;2 are constructed for the first time in the literature. The binary generators of these codes are available online at [7]. In addition to these, some known codes are reconstructed via this extension. The results are tabulated.