Consider all possible 3bit datawords totally 8 Recall a pari

Consider all possible 3-bit datawords (totally 8). Recall a parity bit is a bit added to a string of binary code that indicates whether the number of 1-bits in the string is even or odd. You can use either even parity bit or odd parity bit.

a.) What is the minimum Hamming distance for the code set you obtained?
b.) How many bit errors can this codeset detect?
c.) What is the number of legal codewords?
d.) What is the number of illegal codewords?

Solution

Answer:

The simplest way of detecting a single error is called a parity bit. In this , the exclusive OR of all the bits of the word.

Consider a word with four bits in total , P , C , B ,A. here P is the parity bit and the other three bits are A,B,C.

For defining the parity P ac C B A , the legal combination is :

P         C         B         A

0          0          0          0

1          0          0          1

1          0          1          0

0          0          1          1

1          1          0          0

0          1          0          1

0          1          1          0

1          1          1          1

The parity bit is called an even parity because it ensures that the total number of 1s in the codeword is even. For the purpose of parity generation, number zero is defined as even.

In the above code set parity bit P is zero : even parity , P is one : odd parity

We have only eight legal combinations out of 16 possible combinations of four bits , because one bit is no longer variable. It’s a function of the other three bits. Here we have taken a 3-bit source word C,B,A and converted into a 4-bit codeword   P,C,B,A.

For example the original source word is 001. The parity bit is 0 0 1 = 1 and the code word is therefore 1001.

The minimum Hamming distance for the code set is 2

Explanation:

If you have two n-bit words X and Y , the hamming distance between X and Y is the number of positions in which the bits of X and Y doffer. The minimum hamming distance is 0 (if X and Y are the same) and the maximum hamming distance is n ( if X is the complement of Y).

In the above code set if we take  

X = 0 0 0 0

Y = 1 0 0 1                Hamming distance = 2

For detecting bit errors in this codeset we have formula     d=s+1

d=2 (hamming distance)

therefore 2 = s+1  

            s = 2-1 = 1

therefore number of bit errors detect in this code set is 1.

Number of legal code words are 4

Explanation : in the above code set even parity are legal code words.

Number of illegal code words are 4

      Explanation : in the above code set odd parity are illegal code words.