First of all, there exist two types of problem.

1. Error detection.    u -->  x --> y -->y = x ?  with high probability

just need know whether the error happens or not 

2.Error correction.   u--> x --> y --> =u with high probability

not only need search the error, but also correct it. 

Begin with q-ary symmetric channel.

Here q is large compared with n => q =

the cardinality of x => {0,1,2,3...q-1] 

Information Theory => average case.(small Pe)

Coding Theory => worst case (Pe = 0)

 you transmit x to y

you have the probability     (no error at i th bit) 

If an error happened in the transmission ,

Information Theory

\     m-l bits                  / \  l bits           /

in the m bits of code, there exist m-l bits of information and l bits of parity code each symbol is a n bits vector.

We need l=o(m) (l<<m) and is small

let l= 2log n, then

Prove that with probability (1-), an error will be detected.

d 

Information Theory 

Ex2 If q is sufficient large prove Pr(detect error) =>1

Coding Theory

How to decode the code

the distance of x and x' : . To correct the error we need

Step 

1.   hamming distance : the number of locations in which

2. Given a code {X}

          min-distance   where x and x'  

3.correct errors the minimum-distance

to detect error we just need np+1

=> singleton bound  (the min distance =d, If we delete d-1 bits, in the remaining parts, the codes still are unique )

      n-d+1  delete d-1

nR =n-d+1

Coding theory

With probability (1-p), an error will be detected.

With probability (1-2p), an error will be corrected.

Repetition code 

for k =1 => n , R=k/n =1/n

error decode : only over n/2 bits have errors.

cons: rate of the code is too low.

hamming code

For example

n=7 k=4 d=3

in each circle, the number of '1' should be even.

the max error corrected = 1

error example

we have

1.a codes [n1 n2 n3 n4 d1 d2 d3] = [1111011]

only in the circle 1( contain d1), the number of '1' =odd number => d1 is an error