15-859V: Introduction to coding theory Spring 2010
Carnegie Mellon University Venkatesan Guruswami
PROB LE M SET 3
Due by Friday, April 23
INSTRUCTIONS
•This problem sets can be turned in groups of two people; i.e., a single write-up for each two person
team suffices. If you prefer, you can also work alone (see the last bullet item for some “credit” for
doing so). Solutions typeset in L
A
T
E
X are preferred.
•You are strongly urged to try and solve the problems without consulting any reference material other
than the course notes and what we cover in class. If for some reason you feel the need to consult
some source (such as a textbook, research paper, or sources on the web), you must acknowledge the
source and also articulate the difficulty you couldn’t overcome before consulting the source and how
it helped you overcome that difficulty. Alternatively, before consulting any such material, I encourage
you to ask me for a hint, preferably by posting a comment on the blog post dedicated to this problem
set, so all students can take advantage of any hints.
•Please use the comments section of the blog for any questions or clarifications about the problems.
•Please start work on the problem set early. The problem set has seven problems and is worth a total of
100 points. As a rough estimate, a score of 70 (or 60 if you work and turn in solutions solo), suffices
for an A on this problem set.
1. (12 points) In this exercise, you are asked to show that good codes can be used to construct expanders
of logarithmic degree.
Let G∈Fn×k
2be the generator matrix of an [n, k]2binary linear code with the property that every
nonzero codeword of Chas Hamming weight in the range [(1/2−)n, (1/2 + )n](such a code is
usually called -biased). Define a graph H= (Fk
2, E)where (x, y )∈Eif x+yequals one of the
nrows of G(if some row occurs multiple times, place an edge of appropriate multiplicity between
x, y).
Prove that the second largest eigenvalue of the adjacency matrix of Hin absolute value is at most
2. Use this to deduce the existence of spectral expanders (with second largest eigenvalue in absolute
value bounded away from the degree) of logarithmic degree.
2. (10 points) Consider the binary expander code based on an unbalanced bipartite (n, m, D, γ, D (1 −
))-expander as defined in lecture (i.e., the code whose parity check matrix is the bipartite adjacency
matrix of the expander) for some < 1/20. Recall that in an (n, m, D, γ , D(1 −))-expander, every
subset Sof up to γn nodes on the left has at least D(1 −)|S|neighbors on the right. In this exercise
you are asked to analyze the following parallel iterative decoder.
For clog nrounds (for a constant cchosen large enough), do the following in parallel for
each variable node: If the variable is in at least 2D/3unsatisfied checks, flip its value.
Prove that the above algorithm corrects any pattern of γ(1 −3)nerrors.
