# 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 $$000110011011011$$ .