Consider the binary linear [6, 3]-code C generated by the basis {011100, 100110, 111001} (a) Express...

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Consider the binary linear [6, 3]-code C generated by the basis {011100, 100110, 111001} (a) Express C as a generator matrix G and transform G into standard form (creating equivalent code C'), listing the performed matrix- operations. (b) List the cosets of C and circle a suitable coset leader within each coset. (c) What is the minimum distance d for the code C'? (d) Determine a parity-check matrix for C' and compute the syndrome associated with each coset. (e) How would the message 101 be encoded by C'? (f) Which 3-bit message would the received transmission 001001 be de- coded as, for your choice of coset leaders (from (b))? (g) An arbitrary 3-bit message is encoded by C' into a 6-bit message and transmitted. Supposing each transmitted bit is corrupted indepen- dently with probability 0.05, compute the overall probabilities of 1) an undetectable error occurring, 2) a detectable error occurring, 3) the message being correctly decoded from the original transmis- sion. Compare these probabilities to the respective probabilities for an unprotected 3-bit message. Consider the binary linear [6, 3]-code C generated by the basis {011100, 100110, 111001} (a) Express C as a generator matrix G and transform G into standard form (creating equivalent code C'), listing the performed matrix- operations. (b) List the cosets of C and circle a suitable coset leader within each coset. (c) What is the minimum distance d for the code C'? (d) Determine a parity-check matrix for C' and compute the syndrome associated with each coset. (e) How would the message 101 be encoded by C'? (f) Which 3-bit message would the received transmission 001001 be de- coded as, for your choice of coset leaders (from (b))? (g) An arbitrary 3-bit message is encoded by C' into a 6-bit message and transmitted. Supposing each transmitted bit is corrupted indepen- dently with probability 0.05, compute the overall probabilities of 1) an undetectable error occurring, 2) a detectable error occurring, 3) the message being correctly decoded from the original transmis- sion. Compare these probabilities to the respective probabilities for an unprotected 3-bit message.

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a The first step is to create a generator matrix G for the given code This is done by listing the basis vectors as rows of a matrix The second step is ... View the full answer