We stated the first JPL bound and outlined the approach of Friedman and Tillich to prove it for linear codes by showing that the dual of a code must have small “essential” covering radius and therefore be large. The method was generalized to non-linear codes by Navon and Samorodnitsky; we will mostly follow their presentation for our proof for the linear case.

We finished most of the (self-contained) proof using Fourier analysis. In particular, we proved the following facts about the Fourier spectrum which we will use to wrap up the proof at the beginning of next lecture:

- If is a function on the hypercube and is the adjacency matrix of the hypercube, then .
- Let be a function on the hypercube and for , let denote the shifted function . Then for every subset , the Fourier transform of the function satisfies
where is the characteristic function of the set . (We will use this for .)

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