  It is conjectured that the dual variety of every smooth nonlinear subvariety
of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an
expectation known as the duality defect conjecture. This would follow from the
truth of Hartshorne's complete intersection conjecture but nevertheless remains
open for the case of subvarieties of codimension $> 2$. A combinatorial
approach to proving the conjecture in the codimension $2$ case was developed by
Holme, and following this approach Oaland devised an algorithm for proving the
conjecture in the codimension $3$ case for particular $N$. This combinatorial
approach gives a potential method of proving the duality defect conjecture in
many of the cases by studying the positivity of certain homogeneous integer
linear recurrence sequences. We give a generalization of the algorithm of
Oaland to the higher codimension cases, obtaining with this bounds the degrees
of counterexamples would have to satisfy, and using the relationship with
recurrence sequences we prove that the conjecture holds in the codimension $3$
case when $N$ is odd.
