v=n,Definition: A $t-(v,k,\lambda)$ orthogonal array ($t \leq k$) (also called an orthogonal array of power/strength $t$) is a $λv^t × k$ array whose entries are chosen from a set $X$ with $v$ points such that in every subset of $t$ columns of the array, every $t-$tuple of points of $X$ appears in exactly $λ$ rows. I have used the standard notation for letters.

It seems to me the OP is looking for orthogonal arrays with minimal number of rows over the set $X=\{0,1\}$ with $v=n,\lambda=1,$ and $t=k$ because he wants every $k-$tuple to occur in every possible $k$ positions.

There are literally hundreds of papers and tens of constructions in this area. There are some lower bounds known for existence. I suggest he start with reading the Wikipedia entry and then moving on to tutorial papers of which there are plenty. Happy googling!

Definition: A $t-(v,k,\lambda)$ orthogonal array ($t \leq k$) (also called an orthogonal array of power/strength $t$) is a $λv^t × k$ array whose entries are chosen from a set $X$ with $v$ points such that in every subset of $t$ columns of the array, every $t-$tuple of points of $X$ appears in exactly $λ$ rows. I have used the standard notation for letters.

It seems to me the OP is looking for orthogonal arrays with minimal number of rows over the set $X=\{0,1\}$ with $k=n,\lambda=1,$ and $t=k$ because he wants every $k-$tuple to occur in every possible $k$ positions.

There are literally hundreds of papers and tens of constructions in this area. There are some lower bounds known for existence. I suggest he start with reading the Wikipedia entry and then moving on to tutorial papers of which there are plenty. Happy googling!

Definition: A $t-(v,k,\lambda)$ orthogonal array ($t \leq k$) (also called an orthogonal array of power/strength $t$) is a $λv^t × k$ array whose entries are chosen from a set $X$ with $v$ points such that in every subset of $t$ columns of the array, every $t-$tuple of points of $X$ appears in exactly $λ$ rows. I have used the standard notation for letters.

It seems to me the OP is looking for orthogonal arrays with minimal number of rows over the set $X=\{0,1\}$ with $\lambda=1,$ and $t=k$ because he wants every $k-$tuple to occur in every $k$ positions.

There are literally hundreds of papers and tens of constructions in this area. There are some lower bounds known for existence. I suggest he start with reading the Wikipedia entry and then moving on to tutorial papers of which there are plenty. Happy googling!

v=n,Definition: A $t-(v,k,\lambda)$ orthogonal array ($t \leq k$) (also called an orthogonal array of power/strength $t$) is a $λv^t × k$ array whose entries are chosen from a set $X$ with $v$ points such that in every subset of $t$ columns of the array, every $t-$tuple of points of $X$ appears in exactly $λ$ rows. I have used the standard notation for letters.

It seems to me the OP is looking for orthogonal arrays with minimal number of rows over the set $X=\{0,1\}$ with $v=n,\lambda=1,$ and $t=k$ because he wants every $k-$tuple to occur in every possible $k$ positions.

There are literally hundreds of papers and tens of constructions in this area. There are some lower bounds known for existence. I suggest he start with reading the Wikipedia entry and then moving on to tutorial papers of which there are plenty. Happy googling!