32. In this problem you will investigate the smallest cyclic linear code of length 9 over...

 

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32. In this problem you will investigate the smallest cyclic linear code of length 9 over F2 which contains the code word 100100000. You can use Sage in this problem, just cut and paste your Sage computations into the body of your HW solutions. - (a) Write down the code polynomial g(X) of degree 8 in F2 (] associated to 100100000. Verify that g(x)h(X) = X9 1 in F2[X] for some h(X) = F2[X], and find h(X). Let C = (g(x)) in R9 = F2[X]/(X 1) and note that h(X) is the parity check polynomial for C. (b) What is k = dim(C)? (c) Use polynomial encoding and g(X) to encode the messages 010101 and 100100. (d) Use the parity check polynomial h(X) to check the received words r = 100110011 and r = 100111011 for errors. (e) Use g(X) and h(X) to find a generator matrix G and a parity check matrix H for C. 32. In this problem you will investigate the smallest cyclic linear code of length 9 over F2 which contains the code word 100100000. You can use Sage in this problem, just cut and paste your Sage computations into the body of your HW solutions. - (a) Write down the code polynomial g(X) of degree 8 in F2 (] associated to 100100000. Verify that g(x)h(X) = X9 1 in F2[X] for some h(X) = F2[X], and find h(X). Let C = (g(x)) in R9 = F2[X]/(X 1) and note that h(X) is the parity check polynomial for C. (b) What is k = dim(C)? (c) Use polynomial encoding and g(X) to encode the messages 010101 and 100100. (d) Use the parity check polynomial h(X) to check the received words r = 100110011 and r = 100111011 for errors. (e) Use g(X) and h(X) to find a generator matrix G and a parity check matrix H for C.