New extremal singly even self-dual codes of lengths 6464 and 6666

For lengths $64$ and $66$, we construct extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. We also construct new $40$ inequivalent extremal doubly even self-dual $[64,32,12]$ codes with covering radius $12$ meeting the Delsarte bound.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1706.0169

New extremal binary self-dual codes of length 68 via short kharaghani array over f_2 + uf_2

In this work, new construction methods for self-dual codes are given. The methods use the short Kharaghani array and a variation of it. These are applicable to any commutative Frobenius ring. We apply the constructions over the ring F_2 + uF_2 and self-dual Type I [64, 32, 12]_2-codes with various weight enumerators obtained as Gray images. By the use of an extension theorem for self-dual codes we were able to construct 27 new extremal binary self-dual codes of length 68. The existence of the extremal binary self-dual codes with these weight enumerators was previously unknown.Comment: 10 pages, 5 table

Upper bound of t value of support t-designs of extremal Type III and IV codes

Let C be an extremal Type III or IV code and D_{w} be the support design of C for a weight w. We introduce the two numbers \delta(C) and s(C): \delta(C) is the largest integer t such that, for all wight, D_{w} is a t-design; s(C) denotes the largest integer t such that there exists a w such that D_{w} is a t-design. In the present paper, we consider the possible values of \delta(C) and s(C).Comment: 29 pages. arXiv admin note: text overlap with arXiv:1311.212