Question

Hamming Coding

If we wanted to achieve a maximum of 1 in 10^12 words containing uncorrected errors, and we have an error probability p=0.001, what is the efficiency of this code (where efficiency=k/n.)

Answer 1

In Hamming codes K/n is defiened as coding rate (or can be considered as efficiency of in some sense )

Where : K bit of input frame results n bits of output fram after coding

Block length n = 2^r -1 and Message length K = 2^r-r -1 and r>= 1

Which suggests that the length of code should be less than 1 power of 2

n = 2⁴⁰-1 = 10¹²(approximately)

Hence, we can choose r = 40

K = 2⁴⁰-40-1

Coding rate (efficiency) = (2⁴⁰-40-1 )/2⁴⁰-1

          = 0.999%