On the possibility of group-theoretic description of an equivalence relation connected to the problem of covering subset in finite fields with cosets of linear subspaces

D. S. Sargsyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 53 (2019) no. 1, pp. 23-27

Résumé

Let $F_q^n$ be an $n$-dimensional vector space over a finite field $F_q$. Let $C(F_q^n)$ denote the set of all cosets of linear subspaces in $F_q^n$. Cosets $H_1,H_2,\dots ,H_s$ are called exclusive if $H_i\not \subseteq H_j, 1\leq i$. A permutation $f$ of $C(F_q^n)$ is called a $C$-permutation if for any exclusive cosets $H,H_1,H_2,\dots ,H_s$ such that $H\subseteq H_1\cup H_2\cup \dots \cup H_s$, we have: i) cosets $f(H),f(H_1),f(H_2),\dots ,f(H_s)$ are exclusive; ii) cosets $f^{-1}(H),f^{-1}(H_1),f^{-1}(H_2),\dots ,f^{-1}(H_s)$ are exclusive; iii) $f(H)\subseteq f(H_1)\cup f(H_2)\cup \dots \cup f(H_s)$; iv) $f^{-1}(H)\subseteq f^{-1}(H_1)\cup f^{-1}(H_2)\cup \dots \cup f^{-1}(H_s)$. In this paper we show that the set of all $C$-permutations of $C(F_q^n)$ is the General Semiaffine Group of degree $n$ over $F_q$.

Détail

Keywords: Nite field, coset, covering, linearized disjunctive normal form, general affine group, general semiaffine group.
Mots-clés : bijection