# Introduction to coding theory (CMU, Spr 2010)

## February 5, 2010

### Lecture 8 summary

Filed under: Lecture summary — Venkat Guruswami @ 3:33 pm

We stated the first JPL bound ${R(\delta) \le h(\frac{1}{2} - \sqrt{\delta (1-\delta)})}$ 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:

1. If ${g}$ is a function on the hypercube and ${A}$ is the adjacency matrix of the hypercube, then ${\widehat{Ag}(\alpha) = \hat{g}(\alpha) (n- 2 {\rm wt}(\alpha))}$.
2. Let ${f}$ be a function on the hypercube and for ${z \in {\mathbb F}_2^n}$, let ${f_z}$ denote the shifted function ${f_z(x) = f(z+x)}$. Then for every subset ${S \subseteq {\mathbb F}_2^n}$, the Fourier transform of the function ${F= \frac{1}{2^n} \sum_{z\in S} f_z}$ satisfies

$\displaystyle \hat{F}(\alpha) = \widehat{1_S}(\alpha) \hat{f}(\alpha) \ ,$

where ${1_S}$ is the characteristic function of the set ${S}$. (We will use this for ${S = C^\perp}$.)