6 qubits, k = 5

 0 0 0 0 0 0
 1 0 0 0 0 0

 0 0 0 0 0 1
 1 0 0 0 0 1

 0 0 0 0 1 0
 1 0 0 0 1 0

 0 0 0 0 1 1
 1 0 0 0 1 1

 0 0 0 1 0 0
 1 0 0 1 0 0

 0 0 0 1 0 1
 1 0 0 1 0 1

 0 0 0 1 1 0
 1 0 0 1 1 0

 0 0 0 1 1 1
 1 0 0 1 1 1

 0 0 1 0 0 0
 1 0 1 0 0 0

 0 0 1 0 0 1
 1 0 1 0 0 1

 0 0 1 0 1 0
 1 0 1 0 1 0

 0 0 1 0 1 1
 1 0 1 0 1 1

 0 0 1 1 0 0
 1 0 1 1 0 0

 0 0 1 1 0 1
 1 0 1 1 0 1

 0 0 1 1 1 0
 1 0 1 1 1 0

 0 0 1 1 1 1
 1 0 1 1 1 1

 0 1 0 0 0 0
 1 1 0 0 0 0

 0 1 0 0 0 1
 1 1 0 0 0 1

 0 1 0 0 1 0
 1 1 0 0 1 0

 0 1 0 0 1 1
 1 1 0 0 1 1

 0 1 0 1 0 0
 1 1 0 1 0 0

 0 1 0 1 0 1
 1 1 0 1 0 1

 0 1 0 1 1 0
 1 1 0 1 1 0

 0 1 0 1 1 1
 1 1 0 1 1 1

 0 1 1 0 0 0
 1 1 1 0 0 0

 0 1 1 0 0 1
 1 1 1 0 0 1

 0 1 1 0 1 0
 1 1 1 0 1 0

 0 1 1 0 1 1
 1 1 1 0 1 1

 0 1 1 1 0 0
 1 1 1 1 0 0

 0 1 1 1 0 1
 1 1 1 1 0 1

 0 1 1 1 1 0
 1 1 1 1 1 0

 0 1 1 1 1 1
 1 1 1 1 1 1