On a characterization of some minihypers in PG(t,q) (q=3 or 4) and its applications to error-correcting codes

Abstract

A set F of f points in a finite projective geometry PG(t, q) is an {f, m; t, q}-minihyper if m (≥0) is the largest integer such that all hyperplanes in PG(t, q) contain at least m points in F where t≥2, f≥1 and q is a prime power. Hamada and Deza [9], [11] characterized all {2vα+1+2vβ+1, 2vα+2vβ;t, q}-minihypers for any integers t,q,α and β such that q≥5 and 0≤α<β<t where v l =(q l −1)/(q−1) for any integer l≥0. Recently, Hamada [5], [6] and Hamada, Helleseth and Ytrehus [18] characterized all {2v1+2v2, 2v0+2v1;t, q}-minihypers for the case t≥2 and q ∈ {3, 4}. The purpose of this paper is to characterize all {2vα+1+2vβ+1, 2vα+2vβ;t, q}-minihypers for any integers t,q,α and β such that q ∈ {3, 4}, 0≤α<β<t and β ≠ α+1 using several results in Hamada and Helleseth [12], [13], [14], [16], [17].