hw$\_$lect$14\_2$:

Ameng the sut of all LS$\text{'}$s$,$ those for which the $1$st row and $1$st col are identical are called Standard. However, in some

places $($like in our book$)$ a standard LS is also required to have the letters in "increasing" order, ie$.$ A$,$ B$,$ C$,$ D$.$ For this

hw you may follow the book's definition, because one can always relabel things until the book's criterion is met. It turns

out there are only four standard $44$ LS$\text{'}$s$.$

a$)$ Generate them. Hint: Start with the $1$ st row and $1$ st col set to A$,$ B$,$ C$,$ D$,$ and then generate the other entries while

keeping in mind that no row or col can have more than $1$ occurrence of any letter.

b$)$ One of the four is shown here. Find all LSs $($standard or not, different or not$)$

that is$/$are orthogonal to it$.$ Hint: set the first row of the matrix to $,,,,$ and

use these Greek letters to fill in the remainder of the LS$,$ while always requiring

LS and orthogonality. In problems like this, some trial$-$and$-$error is often required.

But if you follow my hints, there will be less trial$-$and error. It really is like sudoku.

Also, it is possible that a solution does not exist.