On mixed codes with covering radius 1 and minimum distance 2 (Q5901517)

Summary: Let \(R, S\) and \(T\) be finite sets with \(|R|=r, |S|=s\) and \(|T|=t\). A code \(C\subset R\times S\times T\) 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.