Posted by: atri | February 9, 2009

## Lecture 12: Reed-Solomon codes and Communication Complexity

In today’s lecture, we proved that Reed-Solomon codes meet the Singleton bound (and hence are Maximum Distance Separable or MDS). We also proved the following property of an MDS code $C\subseteq \Sigma^n$ with (intergal) dimension $k$: For any subset of indices $S\subseteq [n]$ of size exactly $k$, the code projected onto $S$ is the “trivial” code of distance one: $\Sigma^k$.

The proof of the distance of RS codes can be found in the fall 07 notes on lecture 12.

In the second part of the lecture, we started to talk about 2-party communication complexity. In particular, I encourage you to think of protocols with low communication complexity for the following functions (in all the cases below $x,y\in\{0,1\}^n$):

1. $f_1(x,y)=1$ if and only if $\sum_i x_i\neq \sum_i y_i$ (the sums are over $\mathbb{F}_2$),
2. $f_2(x,y)=1$ if and only if $|x|+|y|\ge t$ (where $|x|$ denotes the Hamming weight of $x$ and $t\ge 1$ is an integer),
3. $f_3(x,y)=1$ if and only if $x>y$ (where we think of $x$ and $y$ as integers),
4. $f_4(x,y)=1$ if and only if $x_i=y_i$ for every $i\in[n]$,
5. $f_5(x,y)=1$ if and only if $x_iy_i=0$ for every $i\in [n]$.

The functions $f_4$ and $f_5$ are known as (set) equality and (set) disjointness in the literature. (The connections to sets comes from the observation that any vector in $\{0,1\}^n$ can be thought of as the “characteristic vector” of a subset of $\{1,\dots,n\}$.

Communication complexity is mostly used a lower bound tool in complexity. For more details, you can take a look at the book with the same name. On Wednesday, we will study some upper bounds on the communication complexity that uses Reed-Solomon codes.