COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

318 A. D. Keedwell
2f 3 = 0 (mod 5), we require that the integers 2, 3 be not adjacent in
either array, and this is clearly impossible. Thus, the non-existence of
orthogonal latin squares of order six appears to be due to a combinatorial
accident.
Our general method for the construction of a pair of mutually orthogonal
latin squares of assigned order I may easily be extended to give a method
for constructing triples. It is easy to see that the latin squares L1 =
(So, Sl, * . >
SI-r}, L,* = {MSo, MSr, . . . , MS,-,}, and G = {M2So,
M2S1, . ..) M2S,-r), where A4 E (I- 1) (0 1 . . . r-2) and SO E 1, Sr,
. . .) S,_, are permutations of the natural numbers 0, 1, . . . , Y - 1, will be
mutually orthogonal provided that the two sets of permutations SJrlMSi
andS;liWSi,i=O,l, . . . . r - 1, are both sharply transitive on the symbolsO,l,
. . . . Y- 1. Since Sz:lMZS, = (S;rA4SJ2, it is clear from Diagram 3
that a sufficient condition for the existence of such a triple of mutually
orthogonal latin squares of order 10 is that a 9X9 matrix A = (a,), i = 1
to 9, j = 0 to 8, exist with the properties:
(i) each of the integers 0, 1, . . . , 9 occurs at most once in each row and
column, and the integer i does not occur in the (i+ l)th column or the
(9 - i)th row ;
(ii) if ali, = a2 j2 = . . . = agj, = r then (a) the integers r+l, alj,+l,
a2 j2+b . . . 2 a9 j,+l are all different (all addition being modulo 9), r = 0,
1,2, . ..) 8, and (b) the integers r+2, arj1+2, asj2+2 . . . , a9js+2 are all different
;
(iii) if arj, = a2je= . . . = agis = 9 then (a) the integers 9, a, jl+I,
a2jz+l, . . a9 a9jp+l are all different, and (b) the integers 9, a, jI+2, a2j2+2,
. . . , a, je+2 are ail different.
To the disappointment of the author, it turns out that the arrays AZ
corresponding to the property D neofields of order 10 have properties (i),
(ii) (a), (iii) (a), and (iii) (b), but fail to satisfy property (ii) (b).
For the purpose of searching for 9X9 matrices of type A, a computer
programme was written which would insert successively the integers alo,
all, . . . , a98 and would backtrack to the preceding place in the event that a
place could not be filled successfully. Details of the construction of this
programme so as to require as few instructions as possible, of the computer
time needed, and of the results appear in [l] and so need not be repeated
here.
S,-lMS,, = (9)(0 1 2 3 4 5 6 7 8)
SilA!fS1 = @)(a10 all al2 al3 al4 al5 a16 al7 ad
S,1MSg = (W90 a91 a92 a93 a94 a95 a96 a97 a981
Diagram 3
Property D neojields and latin squares
REFERENCES
1. A. D. KEEDWELL: On orthogonal latin squares and a class of neofields. Rend. Mat. e
A&. (5) 25 (1966), 519-561.
2. A. D. KEEDWELL: On property Dneofields. Rend. Mat. e AppZ.(5)26(1967), 384-402.
3. L. J. PAIGE: Neofields. Duke Math. J. 16 (1949), 39-60.
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