A binary linear cyclic code Ccyc(n, k) has code length n= 7 and generator poly- nomial...

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A binary linear cyclic code Ccyc(n, k) has code length n= 7 and generator poly- nomial g(x) = 1 + X² + X³ + X4. (a) Find the code rate, the generator and parity check matrices of the code in systematic form, and its Hamming distance. (b) If all the information symbols are '1's, what is the corresponding code vector? (c) Find the syndrome corresponding to an error in the first information symbol, and show that the code is capable of correcting this error. A binary linear cyclic code Ccyc(n, k) has code length n= 7 and generator poly- nomial g(x) = 1 + X² + X³ + X4. (a) Find the code rate, the generator and parity check matrices of the code in systematic form, and its Hamming distance. (b) If all the information symbols are '1's, what is the corresponding code vector? (c) Find the syndrome corresponding to an error in the first information symbol, and show that the code is capable of correcting this error.

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