On Mixed Codes with Covering Radius 1 and Minimum Distance 2

Let R, S and T be finite sets with |R | = r, |S | = s and |T | = t. A code C ⊂ R × S × T with covering radius 1 and minimum distance 2 is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality K(r, s, t; 2). These bounds turn out to be best possible in many instances. Focussing on the special case t = s we determine K(r, s, s; 2) when r divides s, when r = s − 1, when s is large, relative to r, when r is large, relative to s, as well as K(3r, 2r, 2r; 2). Some open problems are posed. Finally, a table with bounds on K(r, s, s; 2) is given.