Mutually Orthogonal Latin Squares and Self-complementary Designs

Suppose that n is even and a set of n/2 -1 mutually orthogonal Latin squares of order n exists. Then we can construct a strongly regular graph with parameters (n², n/2 (n-1), n/2 ( n/2-1), n/2 ( n/2 -1)), which is called a Latin square graph. In this paper, we give a sufficient condition of the Latin square graph for the existence of a projective plane of order n. For the existence of a Latin square graph under the condition, we will introduce and consider a self-complementary 2-design (allowing repeated blocks) with parameters (n, n/2 , n/2 ( n/2 -1)). For n &#8801; 2 (mod 4), we give a proof of the non-existence of the design.</p

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

A note on the Assmus--Mattson theorem for some ternary codes (a resume)

Let $C$ be a two and three-weight ternary code. Furthermore, we assume that $C_\ell$ are $t$-designs for all $\ell$ by the Assmus--Mattson theorem. We show that $t \leq 5$. As a corollary, we provide a new characterization of the (extended) ternary Golay code.Comment: 6 pages, this is a resume of "A note on the Assmus--Mattson theorem for some ternary codes

On the strongly regular graphs obtained from quasi-symmetric 2-(31,7,7) designs

It is known that the five non-isomorphic quasi-symmetric 2-(31, 7, 7) designs lead to non-isomorphic strongly regular graphs with parameters (155, 42, 17, 9). We will show that there exist no isomorphisms among these graphs and the block graphs of the Steiner triple systems STS(31) except the isomorphism between the block graphs of the point-plane design and the point-line design of PG(4, 2)

A note on the Assmus--Mattson theorem for some binary codes II

Let $C$ be a four-weight binary code, which has all one vector. Furthermore, we assume that $C$ supports $t$-designs for all weights obtained from the Assmus--Mattson theorem. We previously showed that $t\leq 5$. In the present paper, we show an analogue of this result in the cases of five and six-weight codes.Comment: 15 pages. arXiv admin note: text overlap with arXiv:2208.0861