Purity testing protocols (PTPs), also known as Bell state testing protocols, i.e., protocols that decide with high probability whether or not a distributed bipartite quantum state is a tensor product of Bell states, have been proven to be a useful tool in many quantum communication applications. In this paper, we provide geometrical constructions for such protocols that originate directly from classical linear error correcting codes (LECCs), in a way that the properties of the resulting PTP are completely determined from those of the LECC used in the construction. We investigate the implications of our results in various quantum communication tasks, including error detection, entanglement purification for general quantum error models and quantum message authentication.

A hypersurface of order $(n+1)(p^h-1)$ in projective space of dimension $n$ of prime characteristic $p$ has an invariant monomial. This implies that a hypersurface of order $(n+1)(p^h-1)-1$ determines an invariant point. A hypersurface of order $d

We consider orthogonal arrays of strength two and even order $q$ having $n$ columns which are equivalent to $n-2$ mutually orthogonal Latin squares of order $q$. We show that such structures induce graphs on $n$ vertices, invariant up to complementation. Previous methods worked only for single Latin squares of even order and were harder to apply. If $q$ is divisible by 4, the invariant graph is simple undirected. If $q$ is 2 modulo 4, the graph is a tournament. When $n=q+1$ is maximal, the array corresponds to an affine plane, and the vertex valencies of the graph have parity $q/2$ modulo 2. We give the graphs at all possible points and lines for 22 planes of order 16. Four of the planes, none of them of translation or dual translation type, produce nonempty graphs at some points.

A formula for Glynn's hyperdeterminant $\det_p$ (p prime) of a square matrix shows that the number of ways to decompose any integral doubly stochastic matrix with row and column sums $p-1$ into $p-1$ permutation matrices with even product, minus the number of ways with odd product, is 1 (mod p). It follows that the number of even Latin squares of order $p-1$ is not equal to the number of odd Latin squares of that order. Thus Rota's basis conjecture is true for a vector space of dimension $p-1$ over any field of characteristic zero or p, and all other characteristics except possibly a finite number. It is also shown where there is a mistake in a published proof that claimed to multiply the known dimensions by powers of two, and that also claimed that the number of even Latin squares is greater than the number of odd Latin squares. Now, 26 is the smallest unknown case where Rota's basis conjecture for vector spaces of even dimension over a field is unsolved.