3. (a) Consider the ternary linear code C of length 5 with parity matrix 0 2...

 

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3. (a) Consider the ternary linear code C of length 5 with parity matrix 0 2 1 1 2 2012 2 0 1 0 0 1 H= (i) What is dim C? (ii) Write out all the elements of C. (iii) Suppose the word y = 11221 is received. Use coset leader decoding to decode y to an element of C. [7 marks] (b) Suppose C is a self-dual linear code with length 10. Prove that dim C = 5. [6 marks] [4 marks] [10 marks] (c) Verify that g(x) = x+x+1 is a factor of 2 + 1 in Z[x]. Hence write down the parity polynomial and a parity matrix for the cyclic code C with canonical generator [7 marks] 3. (a) Consider the ternary linear code C of length 5 with parity matrix 0 2 1 1 2 2012 2 0 1 0 0 1 H= (i) What is dim C? (ii) Write out all the elements of C. (iii) Suppose the word y = 11221 is received. Use coset leader decoding to decode y to an element of C. [7 marks] (b) Suppose C is a self-dual linear code with length 10. Prove that dim C = 5. [6 marks] [4 marks] [10 marks] (c) Verify that g(x) = x+x+1 is a factor of 2 + 1 in Z[x]. Hence write down the parity polynomial and a parity matrix for the cyclic code C with canonical generator [7 marks]