  We give one more proof of the first linear programming bound for binary
codes, following the line of work initiated by Friedman and Tillich. The new
argument is somewhat similar to previous proofs, but we believe it to be both
simpler and more intuitive. Moreover, it provides the following 'geometric'
explanation for the bound. A binary code with minimal distance $\delta n$ is
small because the projections of the characteristic functions of its elements
on the subspace spanned by the Walsh-Fourier characters of weight up to
$\left(\frac 12 - \sqrt{\delta(1-\delta)}\right) \cdot n$ are essentially
independent. Hence the cardinality of the code is bounded by the dimension of
the subspace.
  We present two conjectures, suggested by the new proof, one for linear and
one for general binary codes which, if true, would lead to an improvement of
the first linear programming bound. The conjecture for linear codes is related
to and is influenced by conjectures of H\r{a}stad and of Kalai and Linial. We
verify the conjectures for the (simple) cases of random linear codes and
general random codes.
