On the 2-(25, 5, λ) design of zero-sum 5-sets in the Galois field GF(25)

In this paper we consider the incidence structure ${\mathcal{D}}=({\mathcal{F}},{\mathcal{B}}_5^{0}),$ where ${\mathcal{F}}$ is the Galois field with $25$ elements, and ${\mathcal{B}}_5^{0}$ is the family of all $5$-subsets of $\mathcal F$ whose elements sum up to zero. It is known that ${\mathcal{D}}$ is a $2$-$(25,5,71)$ design. Here we provide two alternative, direct proofs of this result and, moreover, we prove that ${\mathcal{D}}$ is not a $3$-design. Furthermore, if ${\mathcal{B}}_5^{x}$ denotes the family of all $5$-subsets of $\mathcal F$ whose elements sum up to a given element $x \in \mathcal F,$ we also provide an alternative, direct proof that $({\mathcal{F}},{\mathcal{B}}_5^{x})$ is not a $2$-design for $x \neq 0.$ If ${\mathcal B}_{5}^{x,\ast}$ denotes the family of all $5$-subsets of $\mathcal F \!\setminus\! \{0\}$ whose elements sum up to $x,$ this also provides an alternative proof that the incidence structure $({\mathcal{F}}\!\setminus\! \{0\},{\mathcal{B}}_5^{x,\ast})$ is a $1$-$(24,5,r)$ design if and only if $x=0.$