A characterization of some minihypers and its application to linear codes
A set \(F\) of points of \(PG(r,q)\) is called an \(\{ f,m;r,q \}\)-minihyper, if \(| F| =f\) and if \(m=\) min\(\{ | F \cap \Delta| : \Delta \in {\mathcal F}_{r-1} \}\), where \({\mathcal F}_{r-1}\) is the set of projective \((r-1)\)-dimensional subspaces of \(PG(r,q)\). One of the author's main result is the following: Let \(\theta_j=\left(q^{j+1}-1\right)/(q-1)\). If \(F\) is an \(\{ f,\theta_{r-2}+s;r,q \}\)-minihyper with \(f \leq \theta_{r-1}+s\theta_1+q-1\), \(1 \leq s \leq q-1\), \(q \geq 3\), \(r \geq 4\), then \(F\) includes a hyperplane of \(PG(r,q)\). The author also studies the problem to find, for \(1 \leq s \leq q-1\), the minimum value of \(f\) for which an \(\{ f,\theta_{r-2}+s;r,q \}\)-minihyper exists. Results in that direction are applied to provide nonexistence results for \(q\)-ary blockcodes with lengths in certain ranges. The proofs involve the Griesmer bound and blocking sets.
scientific article; zbMATH DE number 1270442