COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

316 A. D. Keedwell
of the elements in corresponding places of any two adjacent secondary
diagonals are constant. Moreover, in each such diagonal, the element
of the pth row and qth column is x times the element of the (pi- 1)th
row and (q- 1)th column, so each element appears exactly once in each
secondary diagonal. Since the equation 1 +x” = 0 is soluble in the field,
one secondary diagonal consists entirely of zeros.
M = (W x . . . x-2)
S;i~MS,-1 = (Y-2)(1 fx’-2 x+X’-* . . . X’-2+X’-2)
s3!L!7r-, = (x’-3)(1+ Y-3 x+x’-3 . . . Y-2+x’-3)
. . . . . . . . . . . . . . . .I. . . .
.*. . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S,lMSz = (x)(1 fx x+x . . . Y-*-J-X8 >
S,lMS1 = (1X1+ 1 x+1 . . . x-2+ 1)
Diagram 1
The above observations lead us to the realization that a sufficient condition
for the existence of a pair of orthogonal latin squares of order P is
that an (r- 1) x (r- 1) matrix A: should exist with the following properties:
(i) the integers 0 to r- 1 appear at most once in each row and column,
and the integer (r- l)- i never occurs in the ith row; (ii) the main secondary
diagonal consists entirely of (r- 1)‘s; (iii) all other secondary diagonals
comprise the elements 0, I,. . .) (r-2) written cyclically; and (iv)
the differences between the elements in corresponding places of each two
adjacent secondary diagonals (excluding the main secondary diagonal)
are all distinct and none is equal to 1. For the details, see [l]. We exhibit
an example of such a matrix for the case r = 10 in Diagram 2.
5 3 0 2 7 6 1 4 9
2 8 1 6 5 0 3 9 4
7 0 5 4 8 2 9 3 1
8 4 3 7 1 9 2 0 6
3 2 6 0 9 1 8 5 7
1 5 8 9 0 7 4 6 2
4 7 9 8 6 3 5 1 0
6 9 7 5 2 4 0 8 3
9 6 4 1 3 8 7 2 5
Diagram 2
A matrix Af having the above properties is completely determined by
its first row and may easily be obtained by computer by successive trial.
Property D neojields and latin squares 317
We require a first row eo, el, . . . , ere3 such that (i) the ei are all different
and are the integers 0, 1,. . . , r - 3 in some order, (ii) the r - 3 differences
di = ei-ei-i are all different (taken modulo (r- 1)) and are the integers
2, 3,. . .) r - 2 in some order, and (iii) the elements e, - i (for i = 0, 1, . . . ,
r- 3) are all different modulo (r- 1) and are the integers 0, 1,. . . , r- 3
in some order.
It is necessary and sufficient for the existence of an array A: that a
property D neofield of order r should exist, and the recognition of this
fact allows the above computer search to be made more economic. A neofield
N is called a property D neofield if (i) its multiplicative group is
cyclic, and (ii) there exists a generator x of N such that (I+ x’)/(l + xt-‘) =
= (1 +x”)/(l +x*-l) implies t = u for all integers t, u (taken modulo
r- 1). The divisibility property (ii) will be referred to as property D. It is
easy to see that the Cayley table of the addition loop of such a neofield,
with first row and column deleted, forms an array Af if we replace powers
of x by their indices and 0 by r- 1. Since 1 + 1 = 0 in a neofield of even
order and 1 +x@-~)‘~ = 0 in a neofield of odd order (see [3]), it follows
that the integer i must not occur in the ith column of an array A: when r is
even and that there is a corresponding restriction when r is odd. Thus,
for example, when r is even, we know that ei + if 1, and this fact reduces
the time required for the search for arrays A: considerably.
A study of property D neofields leads to a number of interesting conjectures.
(i) Do there exist property D neofields of all finite orders r except 6?
Certainly this is true for all r -== 21.
(ii) Can it be proved that both commutative and non-commutative
D-neofields exist for all r > 14 and that the number of isomorphically distinct
D-neofields of assigned order r increases with r?
(iii) Do there exist planar property D neofields which are not fields?
None of those so far obtained by the author are planar either in the sense
of Paige [3] or of Keedwell [l], as is proved in [2].
(iv) Is it true that, if a finite D-neofield of even order has characteristic
2 (or, equivalently, has the inverse property), then it is a field? Is the result
true when the neofields in question are restricted to being commutative?
It remains to explain the non-existence of matrix arrays A: when r = 6.
As explained in detail in [l], the necessary and sufficient condition for the
existence of an array A: (or of a property D neofield of order r) may be
re-formulated as follows : “A necessary and sufficient condition that an
array A: exists for a given integer r is that the residues 2, 3, . . . , r-2,
modulo (r- I), can be arranged in a row array P, in such a way that the
partial sums of the first one, two, . . . , (r - 3), are all distinct and non-zero
modulo (r- 1) and so that, in addition, when each element of the array is
reduced by 1, the new array Pi has the same property.” In the case when
r = 6, P, comprises the integers 2, 3, 4 and Pi comprises 1, 2, 3. Since