This may be enough to show that the growth rate and the conjecture for balanced sets is the same as for general sets.

]]>For k=6 this construction gives a set of size 6×26=156 but using further random trials I now have

]]>{ 0 0 0 0 }

{ 0 3 3 1 }

{ 0 3 3 2 }

{ 1 3 3 0 }

{ 2 3 3 0 }

{ 3 0 3 1 }

{ 3 0 3 2 }

{ 3 1 3 0 }

{ 3 2 3 0 }

{ 3 3 0 1 }

{ 3 3 0 2 }

{ 3 3 1 0 }

{ 3 3 2 0 }

]]>