Characterization of some minihypers in PG(4,3)

In this paper we characterize the minihypers with parameters $(66,21)$ in the geometry PG(4,3). These parameters are important because they are instrumental in solving the problem of the existence of several hypothetical ternary Griesmer codes of dimension $6$. This classification gives also insight into the classification problem for $(v_r+2v_{r-1},v_{r-1}+2v_{r-2})$-mimihypers in PG$(r,q)$ for any $r\ge3$ and any prime power $q\ge4$.