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Register a new userMutually orthogonal binary frequency squares
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A emph{frequency square} is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only {em binary} frequency squares of order $n$ with $n/2$ zeroes and $n/2$ ones in each row and column. Two such frequency squares are emph{orthogonal} if, when superimposed, each of the 4 possible ordered pairs of entries occurs equally often. In this context we say that a $k$-MOFS$(n)$ is a set of $k$ binary frequency squares of order $n$ in which each pair of squares is orthogonal. A $k$-MOFS$(n)$ must satisfy $kle(n-1)^2$, and any MOFS achieving this bound are said to be emph{complete}. For any $n$ for which there exists a Hadamard matrix of order $n$ we show that there exists at least $2^{n^2/4-O(nlog n)}$ isomorphism classes of complete MOFS$(n)$. For $2<nequiv2pmod4$ we show that there exists a $17$-MOFS$(n)$ but no complete MOFS$(n)$. A $k$-maxMOFS$(n)$ is a $k$-MOFS$(n)$ that is not contained in any $(k+1)$-MOFS$(n)$. By computer enumeration, we establish that there exists a $k$-maxMOFS$(6)$ if and only if $kin{1,17}$ or $5le kle 15$. We show that up to isomorphism there is a unique $1$-maxMOFS$(n)$ if $nequiv2pmod4$, whereas no $1$-maxMOFS$(n)$ exists for $nequiv0pmod4$. We also prove that there exists a $5$-maxMOFS$(n)$ for each order $nequiv 2pmod{4}$ where $ngeq 6$.
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A frequency square is a square matrix in which each row and column is a permutation of the same multiset of symbols. A frequency square is of type $(n;lambda)$ if it contains $n/lambda$ symbols, each of which occurs $lambda$ times per row and $lambda$ times per column. In the case when $lambda=n/2$ we refer to the frequency square as binary. A set of $k$-MOFS$(n;lambda)$ is a set of $k$ frequency squares of type $(n;lambda)$ such that when any two of the frequency squares are superimposed, each possible ordered pair occurs equally often. A set of $k$-maxMOFS$(n;lambda)$ is a set of $k$-MOFS$(n;lambda)$ that is not contained in any set of $(k+1)$-MOFS$(n;lambda)$. For even $n$, let $mu(n)$ be the smallest $k$ such that there exists a set of $k$-maxMOFS$(n;n/2)$. It was shown in [Electron. J. Combin. 27(3) (2020), P3.7] that $mu(n)=1$ if $n/2$ is odd and $mu(n)>1$ if $n/2$ is even. Extending this result, we show that if $n/2$ is even, then $mu(n)>2$. Also, we show that whenever $n$ is divisible by a particular function of $k$, there does not exist a set of $k$-maxMOFS$(n;n/2)$ for any $kle k$. In particular, this means that $limsup mu(n)$ is unbounded. Nevertheless we can construct infinite families of maximal binary MOFS of fixed cardinality. More generally, let $q=p^u$ be a prime power and let $p^v$ be the highest power of $p$ that divides $n$. If $0le v-uh<u/2$ for $hge1$ then we show that there exists a set of $(q^h-1)^2/(q-1)$-maxMOFS$(n;n/q)$.
We give a simple construction of an orthogonal basis for the space of m by n matrices with row and column sums equal to zero. This vector space corresponds to the affine space naturally associated with the Birkhoff polytope, contingency tables and Latin squares. We also provide orthogonal bases for the spaces underlying magic squares and Sudoku boards. Our construction combines the outer (i.e., tensor or dyadic) product on vectors with certain rooted, vector-labeled, binary trees. Our bases naturally respect the decomposition of a vector space into centrosymmetric and skew-centrosymmetric pieces; the bases can be easily modified to respect the usual matrix symmetry and skew-symmetry as well.
We study the connection between mutually unbiased bases and mutually orthogonal extraordinary supersquares, a wider class of squares which does not contain only the Latin squares. We show that there are four types of complete sets of mutually orthogonal extraordinary supersquares for the dimension $d=8$. We introduce the concept of physical striation and show that this is equivalent to the extraordinary supersquare. The general algorithm for obtaining the mutually unbiased bases and the physical striations is constructed and it is shown that the complete set of mutually unbiased physical striations is equivalent to the complete set of mutually orthogonal extraordinary supersquares. We apply the algorithm to two examples: one for two-qubit systems ($d=4$) and one for three-qubit systems ($d=8$), by using the Type II complete sets of mutually orthogonal extraordinary supersquares of order 8.
We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse holds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a prime power. This corroborates further the general belief that a complete collection of MUBs can only exist in prime power dimensions. The connection to ODs of sl_n(C) potentially allows the application of known results on (partial) ODs of sl_n(C) to MUBs.
We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2x3 pattern is 1/720+o(1). This result is the best possible in the sense that 2x3 cannot be replaced with 2x2 or 1xN for any N.
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