It is unknown whether or not there exists a \([51, 5, 33; 3]\)-code (meeting the Griesmer bound). The purpose of this paper is to show that there is no \([51, 5, 33; 3]\)-code.

The Hitting Set problem is investigated in relation to restrictions imposed on the cardinality of subsets and the frequency of element occurences in the subsets. It is shown that the Hitting Set subproblem where each subset has cardinality \(C\) for fixed \(C \geq 2\) and the frequency of each element is exactly \(f\) for fixed \(f \geq 3\) remains NP-complete, but the problem becomes polynomial when \(f \leq 2\). The restriction of the Vertex Cover problem to \(f\)-regular graphs for \(f \geq 3\) remains NP-complete.

Hill and Newton showed that there exists a \([20, 6, 12; 3]\)-code, and that the weight distribution of a \([20,5, 12; 3]\)-code is unique. However, it is unknown whether or not a code with these parameters is unique. Recently, Hamada and Helleseth showed that a \([19, 4, 12; 3]\)-code is unique up to equivalence, and characterized this code using a characterization of \(\{21, 6; 3, 3\}\)-minihypers. The purpose of this paper is to show, using the geometrical structure of the \([19, 4, 12; 3]\)-code, that exactly two non-isomorphic \([20, 5, 12; 3]\)-codes exist.