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In 1998, Khodkar showed that the minimal critical set in the Latin
square corresponding to the elementary abelian $2$-group of order
16 is of size at most 124. Since the paper was published, improved
methods for solving integer programming problems have been
developed. Here we give an example of a critical set of size 121
in this Latin square, found through such methods. We also give a
new upper bound on the size of critical sets of minimal size for
the elementary abelian $2$-group of order $2^n: 4^n - 3^n + 4 -
2^n - 2^{n-2}$. We speculate about possible lower bounds for this
value, given some other results for the elementary abelian
$2$-group of orders 32 and 64. An example of a critical set of
size 29 in the Latin square corresponding to the elementary
abelian $3$-group of order 9 is given, and it is shown that any
such critical set must be of size at least 24, improving the bound
of 21 given by Donovan, Cooper, Nott and Seberry.
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