# HW Number 1

## Due on February 25th

- Let \(\mathcal{C} = \{ 01010, 10101 \}\) be a binary code transmitted through a binary symmetric channel with flip probability of \(p = .1\). Using a nearest codeword decoder, what is the probability of incorrectly decoding a word?
- Suppose \(a, b, c\) are words in \(\mathbb{F}_2^n\) (all binary strings of length \(n\)) such that
$$ d(a,b) = d(a,c) = d(b,c) = 2k $$
Show that there is exactly one word \(x\) such that \(d(a,x) = d(b,x) = d(c,x) = k\).
- Using a standard Hamming Encoding of length 15 (a parity bit at 1, 2, 4, and 8) encode the message \(11111100000\). Now decode (there is \(\leq 1\) error) the message \(000110011011011\) .