Descriptions

  associative: Elements symmetrically opposite from the center are a complementary pair, i.e., their sum is n-1, (for element range 0 to n-1).

  near associative: Associative with the exception of 2 pairs of elements. For singly even orders.

  axial symmetric: One-to-one correspondence between all opposite pairs of elements in each row or in each column.

  double axial symmetric: One-to-one correspondence between all opposite pairs of elements in each row and in each column.

  center symmetric: One-to-one correspondence between all elements symmetrically opposite from the center.

  near center symmetric: Center symmetric with the exception of 2 pairs of elements. For singly even orders.

  pandiagonal, (PLS): Knut Vik design.

  weakly pandiagonal, (WPLS): The sum of the n numbers in each of the 2n diagonals is the same.

  nfr: natural order first row, 0 1 2 ..

  nfc: natural order first column, 0 1 2 ..

  nfr nfc: nfr and nfc, (reduced LS).

  nbd: natural order \diagonal, 0 1 2 ..

  self-transpose: LS is equal to its transpose, (symmetric matrix).

LatinSquares Program Notes

Uses backtracking to make small order Latin squares, LS, and diagonal Latin squares, DLS, of various types:

  LS: LS, axial symmetric, double axial symmetric, center symmetric, orthogonal, self-orthogonal,
       natural order first row and first column, self-transpose

  DLS: DLS, axial symmetric, double axial symmetric, center symmetric, orthogonal, self-orthogonal, associative

Here LS exclude DLS, and axial symmetric exclude double axial symmetric.

All squares, except associative, are made with natural order first row, (NFR), 0 1 2 ... n-1.
Associative are natural order \diagonal. For DLS, the center symmetric squares equate to the NFR permutation of the associative squares.

Symmetric means that there is one-to-one correspondence between all opposite pairs of elements. Axial symmetric means opposite in each row or in each column. Double axial symmetric means opposite in each row and in each column.

Orthogonal refers to two adjacent squares.