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

Understand

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.