All linear block codes can be defined using a parity$-$check matrix. In

other words, the operation of any linear block code can be described

using a Tanner graph that connects a set of bit nodes to another set

of check nodes.

In this question, assume transmission over a binary symmetric

channel $($BSC$)$ with an error probability $p.$

Also assume that the bit$-$flipping algorithm is used to decode a

particular linear block code over this BSC$.$

We have seen in class that the bit$-$flipping algorithm is going to work

well provided that each check node in the Tanner graph of this linear

block code is fed with, at most, one wrong bit.

Consider a particular check node connected to $l$ bit nodes $c0,c1,$

$c2,$dots, and $cl-1.$

Find the expression of the probability that this check node

receives two or more wrong bits.

Simplify the equation thus obtained by assuming that $p1.$

By using the result obtained in Question $1,$ determine a condition that must be satisfied

for successful decoding of a linear block code using the bit$-$flipping algorithm.