Consider the association scheme of the n-cube,
with vertex set X={0,1}^n and relation R_i={(x,y): d(x,y)=i}
where d denotes Hamming distance. The adjacency matrix of the
n-cube generates a Bose-Mesner algebra, and if we extend this
algebra by the diagonal matrix with (x,x)-entry n-2d(0,x), we
obtain the Terwilliger algebra of the n-cube. This can also
be viewed as the algebra of matrices that commute with the
coordinatewise action of the symmetric group S_n on X.

A few years ago, the effort to find upper bounds for the size
of a binary error-correcting code seemed to come to an impasse.
A. Samorodnitsky proved that the optimum of Delsarte's linear
program could never be equal to (or even close to, asymtotically)
the Gilbert-Varshamov lower bound. The present talk is one of
several efforts to bring new tools to the task of deriving
upper bounds. (A closely related result, due to A. Schrijver,
made headlines in 2004.) We apply some basic representation
theory to describe all of the extreme rays of the positive
semidefinite cone of the Terwilliger algebra. These are in
one-to-one correspondence with all irreducible S_n-submodules
of the module {R}^X. As a result, we obtain explicit
linear inequalities which can be appended to Delsarte's system
of inequalities (the ``linear programming method'') to improve
overall upper bounds.

We illustrate our technique by extending work of de Klerk and
Paschnik. They applied semidefinite programming to bound the
size of a subset C of X having d(x,y) \neq n/2 for all
x,y in C. Our approach is an iterative one, beginning with
Delsarte's linear programming problem and adding inequalities
based on a Lagrange multiplier analysis of the current optimal
solution.

Based on joint work with Terry Visentin, University of Winnipeg.