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A 2- (v , k , λ) block design (P , B) is additive if, up to isomorphism, P can be represented as a subset of a commutative group (G , +) in such a way that the k elements of each block in B sum up to zero in G. If, for some suitable G , the embedding of P in G is also such that, conversely, any zero-sum k -subset of P is a block in B , then (P , B) is said to be strongly additive. In this paper we exhibit the very first examples of additive 2-designs that are not strongly additive, thereby settling an open problem posed in 2019. Our main counterexample is a resolvable 2- (16 , 4 , 2) design (F 4 × F 4 , B 2) , which decomposes into two disjoint isomorphic copies of the affine plane of order four. An essential part of our construction is a (cyclic) decomposition of the point-plane design of AG (4 , 2) into seven disjoint isomorphic copies of the affine plane of order four. This produces, in addition, a solution to Kirkman's schoolgirl problem. [ABSTRACT FROM AUTHOR]

Let G be the additive group of a finite field. J. Li and D. Wan determined the exact number of solutions of the subset sum problem over G , by giving an explicit formula for the number of subsets of G of prescribed size whose elements sum up to a given element of G. They also determined a closed-form expression for the case where the subsets are required to contain only nonzero elements. In this paper we give an alternative proof of the two formulas. Our argument is purely combinatorial, as in the original proof by Li and Wan, but follows a different and somehow more "natural" approach. We also indicate some new connections with coding theory and combinatorial designs. [ABSTRACT FROM AUTHOR]