n:=15;

# Diagonally cyclic Latin square
L:=List([1..n],i->List([1..n],j->0));
for j in [1..n] do L[1][j]:=((2*(j-1)) mod n)+1; od;
for i in [2..n] do for j in [1..n] do L[i][j]:=(L[i-1][((j-2) mod n)+1] mod n)+1; od; od;
L;

# Fixed diagonal Latin square
M:=List([1..n],i->List([1..n],j->((j-i) mod n)+1));
M;

gap> L;
[ [ 1, 3, 5, 7, 9, 11, 13, 15, 2, 4, 6, 8, 10, 12, 14 ],
  [ 15, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13 ],
  [ 14, 1, 3, 5, 7, 9, 11, 13, 15, 2, 4, 6, 8, 10, 12 ],
  [ 13, 15, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11 ],
  [ 12, 14, 1, 3, 5, 7, 9, 11, 13, 15, 2, 4, 6, 8, 10 ],
  [ 11, 13, 15, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9 ],
  [ 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15, 2, 4, 6, 8 ],
  [ 9, 11, 13, 15, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7 ],
  [ 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15, 2, 4, 6 ],
  [ 7, 9, 11, 13, 15, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5 ],
  [ 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15, 2, 4 ],
  [ 5, 7, 9, 11, 13, 15, 2, 4, 6, 8, 10, 12, 14, 1, 3 ],
  [ 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15, 2 ],
  [ 3, 5, 7, 9, 11, 13, 15, 2, 4, 6, 8, 10, 12, 14, 1 ],
  [ 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15 ] ]

gap> M;
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ],
  [ 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ],
  [ 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 ],
  [ 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ],
  [ 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ],
  [ 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ],
  [ 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9 ],
  [ 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8 ],
  [ 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7 ],
  [ 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6 ],
  [ 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5 ],
  [ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4 ],
  [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3 ],
  [ 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2 ],
  [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1 ] ]