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 with (intergal) dimension
: For any subset of indices
of size exactly
, the code projected onto
is the “trivial” code of distance one:
.
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 ):
if and only if
(the sums are over
),
if and only if
(where
denotes the Hamming weight of
and
is an integer),
if and only if
(where we think of
and
as integers),
if and only if
for every
,
if and only if
for every
.
The functions and
are known as (set) equality and (set) disjointness in the literature. (The connections to sets comes from the observation that any vector in
can be thought of as the “characteristic vector” of a subset of
.
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.
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