A new family of additive designs

Abstract

In this paper we construct a family of 2-(q n , sp 2 , λ) additive designs D = (P, B), where q is a power of a prime p and P is a n-dimensional vector space over GF(q), and we compute their parameters explicitly. These designs, except for some special cases, had not been considered in the previous literature on additive block designs.

Subset sums and block designs in a finite vector space

“…which, again by (5), is independent of y ∈ V * , and which trivially coincides with r x, * k (y) in the case where x − iy = 0 for all i = 1, . .…”

“…Remark 3. 5 We noted in the proof of Theorem 3.1 (resp., Theorem 3.4) that the question of whether (V,…”

“…Under the hypotheses of Theorem 3.6, one may ask whether D = (V, B k ) is also a 3-( p d , k, λ 3 ) design. The question appears to be legitimate, since the answer is affirmative in the case where p = 2 and d ≥ 3, for any even k with 4 ≤ k ≤ 2 d − 4 [10, Proposition 2.5] (see also [5,Remark 2.4]). For p odd, however, D is not necessarily a 3-design.…”

“…Finally, note that D = (V, B 5 ) contains, as a minimal 2-subdesign, the point-line design of the affine plane AG (2,5), which is a 2-(25, 5, 1) design with 30 blocks and replication number r = 6. By Theorem 3.6, every automorphism of D is an invertible affine mapping of V over F 5 , hence it induces an automorphism of AG (2,5), and conversely, by the fundamental theorem of affine geometry. This says precisely that D(x) = (V, B x 33 ) satisfies the necessary condition λ(x)(v − 1) = r (x)(k − 1) for being a 2-design.…”

“…for all other values of i, by (5). In view of ( 15), ( 16), (4), and (5), it is necessary to establish the exact interval [np, (n + 1) p) containing k − i, for the only values of i for which x − iy = 0, that is, for the values i = h + lp, with l ∈ {0, 1, .…”