Q1. Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
- i)) If the weight of each element in the generating matrix of a linear code is at least r, the mininum distance of the code is at least r. (80 words)
- ii)) There is no linear self orthogonal code of odd length. (80 words)
- iii)) There is no 3-cyclotomic coset modulo 121 of size 25. (80 words)
- iv)) There is no duadic code of length 15 over F2. (80 words)
- v)) There is no LDPC code with parameters n = 16, c = 3 and r = 5. (80 words)
Key Points:
- Min distance: smallest weight of any non-zero codeword.
- Self-orthogonal binary codes: all codewords have even weight.
- Cyclotomic coset size: divides Euler's totient function of `n/gcd(s,n)`.
- Binary duadic codes: exist for any odd length `n > 1`.
Answer: