Minihypers and linear codes meeting the Griesmer bound: Improvements to results of Hamada, Helleseth and Maekawa
This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound. In [10,12], it was shown that any {Sigma(i=0)(t-1)epsilon(i)v(i)+1, Sigma(i=0)(t-1)epsilon(i)v(i); t, q}-minihyper, with Sigma(i=0)(t-1)epsilon(i) = h, where (h - 1)(2)   3, q non-square, to h less than or equal to q(6/9)/(1 + q(1/9)), (2) for q = p(f) non-square, p = 2, 3, to h less than or equal to 2(-1/3)q(5/9), (3) for q = p(f) square, p = 2, 3, to h less than or equal to min{2rootq - 1, 2(-1/3)q(5/9)}, and (4) for q = p(f) square, p prime, p > 3, to h less than or equal to min{2rootq - 1, q(6/9)/(1 + q(1/9))}. In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry PG(l, rootq) of PG(t, q). For the coding-theoretical problem, our results classify the corresponding [n = v(t+1) -Sigma(i=0)(t-1) epsilon(i)v(i+1), k = t + 1, d = q(t) - Sigma(i=0)(t-1)epsilon(i); q] codes meeting the Griesmer bound.
minihypers linear codes griesmer bound projective spaces