Lunch Bunch arrangement problem

[tall_man]

A couple of years ago I worked on an arrangement problem, a "lunch bunch." The basic idea is that 64 people want to go out to lunch together in eight groups of eight at a time, with the groups rearranging each month so that everyone gets to have lunch with each of the others and no two people have lunch together twice.

With 49 people, or other prime squares, we can arrange a square and make the sets as follows:

   1  2  3  4  5  6  7 <- shift by 0
   8  9 10 11 12 13 14 <- shift by 1
  15 16 17 18 19 20 21 <- shift by 2
  22 23 24 25 26 27 28 <- shift by 3
  29 30 31 32 33 34 35 <- shift by 4
  36 37 38 39 40 41 42 <- shift by 5
  43 44 45 46 47 48 49 <- shift by 6

The first set is the rows as-is, the second set is the columns as-is, and the subsequent sets are generated by shifting the rows as indicated and then reading down the columns. This gives eight total groupings, which covers all the members with no duplication.

When I went to expand the solution to 8x8, the simple shifting idea wouldn't work, because shifts by 2, 4, and 6 caused duplicated positions. What I ending up doing was finding a Galois field of order 8 and using that for shifting instead.

# The multiplication and addition tables are derived from
# a Galois field order(2^3), from (0, 1, a, a^2, ... a^6)
# where a^3 = a + 1.
#
# multiply
#         0     1     a    a^2   a^3   a^4   a^5   a^6
# ------------------------------------------------------
#   0  |  0  |  0  |  0  |  0  |  0  |  0  |  0  |  0  |
#      |------------------------------------------------
#   1  |  0  |  1  |  a  | a^2 | a^3 | a^4 | a^5 | a^6 |
#      |------------------------------------------------
#   a  |  0  |  a  | a^2 | a^3 | a^4 | a^5 | a^6 |  1  |
#      |------------------------------------------------
#   a^2|  0  | a^2 | a^3 | a^4 | a^5 | a^6 |  1  |  a  |
#      |------------------------------------------------
#   a^3|  0  | a^3 | a^4 | a^5 | a^6 |  1  |  a  | a^2 |
#      |------------------------------------------------
#   a^4|  0  | a^4 | a^5 | a^6 |  1  |  a  | a^2 | a^3 |
#      |------------------------------------------------
#   a^5|  0  | a^5 | a^6 |  1  |  a  | a^2 | a^3 | a^4 |
#      |------------------------------------------------
#   a^6|  0  | a^6 |  1  |  a  | a^2 | a^3 | a^4 | a^5 |
#      |------------------------------------------------
# 
# add
# 
#         0     1     a    a^2   a^3   a^4   a^5   a^6
# ------------------------------------------------------
#   0  |  0  |  1  |  a  | a^2 | a^3 | a^4 | a^5 | a^6 |
#      |------------------------------------------------
#   1  |  1  |  0  | a^3 | a^6 |  a  | a^5 | a^4 | a^2 |
#      |------------------------------------------------
#   a  |  a  | a^3 |  0  | a^4 |  1  | a^2 | a^6 | a^5 |
#      |------------------------------------------------
#   a^2| a^2 | a^6 | a^4 |  0  | a^5 |  a  | a^3 |  1  |
#      |------------------------------------------------
#   a^3| a^3 |  a  |  1  | a^5 |  0  | a^6 | a^2 | a^4 |
#      |------------------------------------------------
#   a^4| a^4 | a^5 | a^2 |  a  | a^6 |  0  |  1  | a^3 |
#      |------------------------------------------------
#   a^5| a^5 | a^4 | a^6 | a^3 | a^2 |  1  |  0  |  a  |
#      |------------------------------------------------
#   a^6| a^6 | a^2 | a^5 |  1  | a^4 | a^3 |  a  |  0  |
#      |------------------------------------------------

BEGIN {
@Field8::mtable = (
           [0,  0,  0,  0,  0,  0,  0,  0],
           [0,  1,  2,  3,  4,  5,  6,  7],
           [0,  2,  3,  4,  5,  6,  7,  1],
           [0,  3,  4,  5,  6,  7,  1,  2],
           [0,  4,  5,  6,  7,  1,  2,  3],
           [0,  5,  6,  7,  1,  2,  3,  4],
           [0,  6,  7,  1,  2,  3,  4,  5],
           [0,  7,  1,  2,  3,  4,  5,  6]
           );

@Field8::atable = (
           [0,  1,  2,  3,  4,  5,  6,  7],
           [1,  0,  4,  7,  2,  6,  5,  3],
           [2,  4,  0,  5,  1,  3,  7,  6],
           [3,  7,  5,  0,  6,  2,  4,  1],
           [4,  2,  1,  6,  0,  7,  3,  5],
           [5,  6,  3,  2,  7,  0,  1,  4],
           [6,  5,  7,  4,  3,  1,  0,  2],
           [7,  3,  6,  1,  5,  4,  2,  0]
           );
}

I would like to generalize this solution to other cases. I think it's impossible for composite orders like 6 or 10, but all orders that are powers of primes should work.

I see a math package on cpan called Math::Pari and that is has functions for Galois fields, but I am not sure how to use it to generate tables like the ones above. Has anyone worked with Math::Pari or other packages that might help? Thanks.