A classification result on weighted {δ (p³ + 1), δ; 3, p³}‐minihypers

Abstract

Abstract:We classify all fdð p 3 þ 1Þ; d; 3;, for a prime number p 0 ! 7, with excess e p 3 . Such a minihyper is a sum of lines and of possibly one projected subgeometry PGð5; pÞ, or a sum of lines and a minihyper which is a projected subgeometry PGð5; pÞ minus one line. When p is a square, also (possibly projected) Baer subgeometries PGð3; p

Characterization results on weighted minihypers and on linear codes meeting the Griesmer bound

“…For q = 7, this means that also an {q + 1 + 2, 1; 2, q}-minihyper must be discussed. This is a 1-fold blocking set in PG (2,7). By results of Blokhuis [3], such a weighted blocking set is the sum of one line and two points.…”

“…Thus, a non-weighted {f, m; N, q}-minihyper of PG(N, q) is a set F of f points of PG(N, q) such that m is the minimum weight of the hyperplanes. This is the definition of a minihyper given by Hamada and Tamari in [15] and it was generalized to the definition of a weighted minihyper in [7]. Using this expression for d, the Griesmer bound for a linear [n, k, d]-code over F q can be expressed as:…”

A study of (x(q + 1), x; 2, q)-minihypers

“…The reason is that such minihypers give rise to Griesmer codes via the well-known construction of Hamada [5,8]. If F is an (x(q + 1), x; 2, q)-minihyper and s is the maximal multiplicity of a point, then sχ P − F is an (s(q 2 + q + 1) − x(q + 1), s(q + 1) − x; 2, q)-arc.…”

Characterization results on arbitrary non-weighted minihypers and on linear codes meeting the Griesmer bound

“…From Definition 1.1 and this convention, it is now clear that an { f, m; N , q}-minihyper of PG(N , q) is a set F of f points of PG(N , q), satisfying |F ∩ H | ≥ m for every hyperplane H in PG(N , q) and where |F ∩ H | = m occurs at least once. This is exactly the definition of an { f, m; N , q}-minihyper found in [17] (Hamada and Tamari) and it was generalized to the definition of a weighted minihyper in [9].…”