Communication - Chapter 4

Question 1

|Bⁿₖ(x)| =

Answer 1

bⁿₖ(x)

Question 2

Informal hamming ball

Answer 2

all the codewords with hamming distance at most k from x

Question 3

H(P)=

Answer 3

minus the sum from i=1 to m of pᵢlogpᵢ

Question 4

What does shannons noisy encoding theorem tell you

Answer 4

Given a noisy channel with crossover probability p and ε a very small number the theorem says you can construct a code with a low chance of mistake and high transmission rate so long as the code is long enough.

Question 5

Shannons noisy encoding theorem is…

Answer 5

best possible.

Question 6

length of a code C

Answer 6

for c∈C |c| is the length of the codewords

Question 7

size of a code C

Answer 7

the number of codwords in a code C

Question 8

top number of hamming ball denotion

Answer 8

length of the codeword

Question 9

bottom number of hamming ball denotion

Answer 9

minimum hamming distance (radius of ball)

Question 10

number in brackets in hamming ball denotion

Answer 10

the centre of the hamming ball

Question 11

What does shannons noisy encoding theorem produce

Answer 11

There exists a n₀=n₀(p,ε) s.t. if n≥n₀ code of length n with probability of wrong decoding less than epsilon and R(C)≥1-H(p+ε)

Question 12

Hypothesis of shannons noisy encoding theorem

Answer 12

Given any p∈[0,1/2) and ε∈(0,1/2-p]