Abstract
| Lingua originale | English |
|---|---|
| Titolo della pubblicazione ospite | Bollettino di Matematica Pura e applicata |
| Pagine | 75-82 |
| Numero di pagine | 8 |
| Volume | IX |
| Stato di pubblicazione | Published - 2017 |
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On the 2-(25, 5, λ) design of zero-sum 5-sets in the Galois field GF(25). / Pavone, Marco.
Bollettino di Matematica Pura e applicata. Vol. IX 2017. pag. 75-82.Risultato della ricerca: Chapter
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TY - CHAP
T1 - On the 2-(25, 5, λ) design of zero-sum 5-sets in the Galois field GF(25)
AU - Pavone, Marco
PY - 2017
Y1 - 2017
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 $x=0.$
UR - http://hdl.handle.net/10447/287561
UR - http://www.aracneeditrice.it/index.php/pubblicazione.html?item=9788825501933
M3 - Chapter
SN - 978-88-255-0193-3
VL - IX
SP - 75
EP - 82
BT - Bollettino di Matematica Pura e applicata
ER -