One more proof of the first linear programming bound for binary codes and two conjectures

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.