On Uniformly Packed [ n , n – k , 4] Codes over GF( q ) and a Class of Caps in PG( k –1, q )

We determine all uniformly packed [ n , k , 4] 4] codes over GF (2) and we derive a non‐trivial necessary condition for the existence of uniformly packed [ n , k , 4] codes over GF ( q ), where q ≠ 2 is a prime power. This condition allows us to classify uniformly packed [ n , k , 4] codes over GF (4). As a corollary we obtain a necessary condition for the existence of a projective ( n , k , h 1 , h 2 ) set S in PG ( k –l, q ) with the property that no three points of S are collinear. A further corollary is a necessary condition for the linear representation of partial quadrangles.