ON THE POSSIBILITY OF GROUP-THEORETIC DESCRIPTION OF AN EQUIVALENCE RELATION CONNECTED TO THE PROBLEM OF COVERING SUBSETS IN FINITE FIELDS WITH COSETS OF LINEAR SUBSPACES

Abstract

Let $ F^{n}_{q} $ be an $ n $-dimensional vector space over a finite field $ F_q $ . Let $ C(F^{n}_{q} ) $ denote the set of all cosets of linear subspaces in $ F^{n}_{q} $. Cosets $ H_1, H_2, \ldots H_s $ are called exclusive if $ H_i \nsubseteq H_j $, $ 1 \mathclose{\leq} i \mathclose{<} j \mathclose{\leq} s $. A permutation $ f $ of $ C(F^{n}_{q} ) $ is called a $ C $-permutation, if for any exclusive cosets $ H, H_1, H_2, \ldots H_s $ such that $ H \subseteq H_1 \cup H_2 \cup \cdots \cup H_s $ we have:
i) cosets $ f(H), f(H_1), f(H_2), \ldots, f(H_s) $ are exclusive;
ii) cosets $ f^{-1}(H), f^{-1}(H_1), f^{-1}(H_2), \ldots, f^{-1}(H_s) $ are exclusive;
iii) $ f(H) \subseteq f(H_1) \cup f(H_2) \cup \cdots \cup f(H_s) $;
vi) $ f^{-1}(H) \subseteq f^{-1}(H_1) \cup f^{-1}(H_2) \cup \cdots \cup f^{-1}(H_s) $.
In this paper we show that the set of all $ C $-permutations of $ C(F^{n}_{q} ) $ is the General Semiaffine Group of degree $ n $ over $ F_q $.