Prove that these conditions are sufficient. That is, prove that any set of parity bits that satisfy conditions (a) and (b) ensure that the resulting code has minimum distance 3. 1.Define 4 parity...

Prove that these conditions are sufficient. That is, prove that any set of parity bits that satisfy conditions (a) and (b) ensure that the resulting code has minimum distance 3.

1.Define 4 parity bits for 11-bit messages that satisfy conditions (a) and (b) from Exercise 4.25.

2. Define 5 parity bits for 26-bit messages that satisfy conditions (a) and (b) from Exercise 4.25.

3. Let ℓ ∈ Z>0 , and let n := 2ℓ − 1. Prove that a code with n-bit codewords, minimum distance 3, and messages of length n − ℓ is achievable. (Hint: look at all ℓ-bit bitstrings; use the bits to identify which message bits are part of which parity bits.

Exercise 4.25

o achieve minimum distance 3, it will suffice to have parity bits with the following properties:

(a) each bit of the original message appears in at least two parity bits.

(b) no two bits of the original message appear in exactly the same set of parity bits