The Spectrum of $r_n(t)$-orthogonalities, for $n \leq 6$

Given a pair $L_1,L_2$ of latin squares of order $n$, let $r=N(L_1,L_2)$ denote the number of distinct ordered pairs which occur when $L_1$ and $L_2$ are superimposed. Consider now a set of $t$ latin squares of order $n$, and denote by $r_n(t)$ the number of distinct pairs which arise when each possible pair of distinct squares is checked to determine their level of $r$-orthogonality. More details and examples can be found here.

The spectrum for latin squares of order $n$ is the set of all positive values that can occur as $r_n(t)$ orthogonalities.


$r$-orthogonalities of sets of $2$ latin squares of order $n$

The spectrum of $r$-orthogonalities of sets of $2$ latin squares is given by the following theorem.

For a positive integer $n$, a pair of $r$-orthogonal latin squares of order $n$ exists if and only if $r \in \left\{ n,n^2 \right\}$ or $n+2 \leq r \leq n^2 - 2$, except when:
  1. $n=2$ and $r=4$
  2. $n=3$ and $r \in \left\{ 5,6,7 \right\}$
  3. $n=4$ and $r \in \left\{ 7,10,11,13,14 \right\}$
  4. $n=5$ and $r \in \left\{ 8,9,20,22,23 \right\}$
  5. $n=6$ and $r \in \left\{ 33,36 \right\}$

$r_n(t)$-orthogonalities of sets of $t$ latin squares of order $n$

The spectrums of $r_n(t)$-orthogonalities for $n \leq 6$ are the following, note that when we say $x - y$ in a spectrum we mean that $x$, $y$ and every value in between are included in the spectrum:

  1. $r_2(2)=2$
  2. $r_3(2)=3,9$
  3. $r_4(2)=4,6,8,9,12,16$
  4. $r_4(3)=12, 16, 20, 21, 22, 26, 27, 28, 30, 32, 33, 34, 36, 40, 48$
  5. $r_5(2)=5, 7, 10 - 19, 21, 25$
  6. $r_5(3)=15, 19, 24, 25, 27, 29 - 59, 63, 75$
  7. $r_5(4)= 30, 36, 38 43, 45, 46, 48 - 120, 122, 124, 126, 130, 132, 150$
  8. $r_6(2)=6, 8 - 32, 34$
  9. $r_6(3)=18, 22, 24 - 94, 96$
  10. $r_6(4)=36, 39, 42, 44 - 184, 188$
  11. $r_6(5)=60, 68, 72, 74, 75, 76, 77, 78, 80 - 298, 300$
To obtain an example of a set of $t$ latin squares of order $n$ with $r_n(t)$-orthogonality choose the approriate values below.


Select the $n$:
Select the $t$:
Select the $r_n(t)$-orthogonality

Frequency of each $r_n(t)$ orthogonality

The following tables show how many sets of $t$ latin squares of order $n$ produce each of the possible $r_n(t)$-orthogonalities.
\[ \begin{array}{lcr} \textbf{n=3} \mbox{ and } \textbf{t=2} \\ r_3(2) & \mbox{number of sets} \\ 3 & 1 \\ 9 & 2 \end{array}\] \[ \begin{array}{lcr} \textbf{n=4} \mbox{ and } \textbf{t=2} \\ r_4(2) & \mbox{number of sets} \\ 4 & 4 \\ 6 & 12 \\ 8 & 6 \\ 9 & 24 \\ 12 & 48 \\ 16 & 2\end{array}\] \[ \begin{array}{lcr} \textbf{n=4} \mbox{ and } \textbf{t=3} \\ r_4(3) & \mbox{number of sets} \\ 12 & 4 \\ 16 & 36 \\ 20 & 54 \\ 21 & 72 \\ 22 & 72 \\ 26 & 144 \\ 27 & 192 \\ 28 & 144 \\ 30 & 576 \\ 32 & 198 \\ 33 & 504 \\ 34 & 72 \\ 36 & 198 \\ 40 & 36 \\ 48 & 2 \end{array}\] \[ \begin{array}{lcr} \textbf{n=5} \mbox{ and } \textbf{t=2} \\ r_5(2) & \mbox{number of sets} \\ 5 & 56 \\ 7 & 200 \\ 10 & 600 \\ 11 & 1200 \\ 12 & 1200 \\ 13 & 6150 \\ 14 & 5400 \\ 15 & 14640 \\ 16 & 14400 \\ 17 & 20050 \\ 18 & 6400 \\ 19 & 4800 \\ 21 & 150 \\ 25 & 18 \end{array}\] \[ \begin{array}{lcr} \textbf{n=5} \mbox{ and } \textbf{t=3} \\ r_5(3) & \mbox{number of sets} \\ 15 & 56 \\ 19 & 600 \\ 24 & 1800 \\ 25 & 1800 \\ 27 & 5400 \\ 29 & 12600 \\ 30 & 1200 \\ 31 & 36450 \\ 32 & 12600 \\ 33 & 42600 \\ 34 & 72000 \\ 35 & 137520 \\ 36 & 141000 \\ 37 & 261000 \\ 38 & 486000 \\ 39 & 721050 \\ 40 & 894600 \\ 41 & 1790400 \\ 42 & 2656800 \\ 43 & 3904500 \\ 44 & 5434200 \\ 45 & 8391640 \\ 46 & 9432000 \\ 47 & 12768750 \\ 48 & 12945000 \\ 49 & 13424400 \\ 50 & 9972000 \\ 51 & 8275500 \\ 52 & 4489200 \\ 53 & 2820600 \\ 54 & 1156800 \\ 55 & 582714 \\ 56 & 144000 \\ 57 & 97200 \\ 58 & 7200 \\ 59 & 27900 \\ 63 & 5700 \\ 75 & 36 \end{array}\]