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

Risultato della ricerca: Chapter

### Abstract

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 $x=0.$
Lingua originale English Bollettino di Matematica Pura e applicata 75-82 8 IX Published - 2017

Sumsets
Zero-sum
Galois field
Subset
Alternatives
Incidence
F-structure
Denote
If and only if
Design
Zero
Family

### Cita questo

Pavone, M. (2017). On the 2-(25, 5, λ) design of zero-sum 5-sets in the Galois field GF(25). In Bollettino di Matematica Pura e applicata (Vol. IX, pagg. 75-82)
Bollettino di Matematica Pura e applicata. Vol. IX 2017. pag. 75-82.

Risultato della ricerca: Chapter

Pavone, M 2017, On the 2-(25, 5, λ) design of zero-sum 5-sets in the Galois field GF(25). in Bollettino di Matematica Pura e applicata. vol. IX, pagg. 75-82.
Pavone M. On the 2-(25, 5, λ) design of zero-sum 5-sets in the Galois field GF(25). In Bollettino di Matematica Pura e applicata. Vol. IX. 2017. pag. 75-82
Pavone, Marco. / On the 2-(25, 5, λ) design of zero-sum 5-sets in the Galois field GF(25). Bollettino di Matematica Pura e applicata. Vol. IX 2017. pagg. 75-82
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T1 - On the 2-(25, 5, λ) design of zero-sum 5-sets in the Galois field GF(25)

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N2 - 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 $x=0.$

AB - 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 $x=0.$

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