JNTU B.Tech III ERROR CORRECTING CODES Previous Question Paper (September - 2021 - R18)

The entropy H(X) is defined as the sum of p(x) log₂(1/p(x)) for all x where p(x) > 0. Given the probabilities: P(1)=0.5, P(2)=0.25, P(5)=0.125, and P(6)=0.125.

Calculating the terms:

• 0.5 log₂(0.5) = 0.5(-1) = -0.5
• 0.25 log₂(0.25) = 0.25(-2) = -0.5
• 0.125 log₂(0.125) = 0.125(-3) = -0.375
• 0.125 log₂(0.125) = 0.125(-3) = -0.375

Summing these gives -(-0.5 - 0.5 - 0.375 - 0.375) = 1.75 bits per symbol.

Let C be a linear block code. If C contains at least one odd-weight codeword, there must exist a codeword w of odd weight. We define a mapping f: C -> C such that f(c) = c + w, where addition is performed modulo 2. Since C is a linear code, if c is a codeword, then c + w is also a codeword.

This mapping is a bijection because it is its own inverse: f(f(c)) = (c + w) + w = c + (w + w) = c + 0 = c. Furthermore, the weight of f(c) is given by wt(c + w). If c is an even-weight codeword, then wt(c + w) is odd because the sum of an even and an odd number is odd. Conversely, if c is an odd-weight codeword, then wt(c + w) is even because the sum of two odd numbers is even.

Thus, the mapping f provides a one-to-one correspondence between the set of even-weight codewords and the set of odd-weight codewords. Consequently, the number of even-weight codewords must be exactly equal to the number of odd-weight codewords.