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 {2v α+1+2v β+1, 2v α+2v β;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 {2v α+1+2v β+1, 2v α+2v β;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].
Partially supported by Grant-in-aid for Scientific Research of the Ministry of Education, Science and Culture under Contract Numbers 403-4005-02640182.
Partially supported by the Scandinavia Japan Sasakawa Foundation. Partially supported by the Norwegian Research Council for Science and the Humanities
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Hamada, N., Helleseth, T. (1992). On a characterization of some minihypers in PG(t,q) (q=3 or 4) and its applications to error-correcting codes. In: Stichtenoth, H., Tsfasman, M.A. (eds) Coding Theory and Algebraic Geometry. Lecture Notes in Mathematics, vol 1518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087992
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