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}.)

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