Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 133(c)( 1 0 0 1 10 1 0 1 1)(d)⎛⎜⎝0 0 0 1 1 1 10 1 1 0 0 1 11 0 1 0 1 0 10 1 1 0 0 1 1⎞⎟⎠8. Construct a (5, 2)-block code. Discuss the error-detection and error-correctioncapabilities of your code.9. Let C be the code obtained from the null space of the matrix⎛0 1 0 0 1⎞H = ⎝ 1 0 1 0 1 ⎠ .0 0 1 1 1Decode the messageif possible.01111 10101 01110 0001110. Suppose that a 1000-bit binary message is transmitted. Assume that theprobability of a single error is p and that the errors occurring in differentbits are independent of one another. If p = 0.01, what is the probability ofmore than one error occurring? What is the probability of exactly two errorsoccurring? Repeat this problem for p = 0.0001.11. Which matrices are canonical parity-check matrices? For those matrices thatare canonical parity-check matrices, what are the corresponding standardgenerator matrices? What are the error-detection and error-correction capabilitiesof the code generated by each of these matrices?(a)⎛⎜⎝1 1 0 0 00 0 1 0 00 0 0 1 01 0 0 0 1⎞⎟⎠(b)⎛⎜⎝0 1 1 0 0 01 1 0 1 0 00 1 0 0 1 01 1 0 0 0 1⎞⎟⎠(c)( 1 1 1 01 0 0 1)(d)⎛⎜⎝0 0 0 1 0 0 00 1 1 0 1 0 01 0 1 0 0 1 00 1 1 0 0 0 1⎞⎟⎠12. List all possible syndromes for the codes generated by each of the matricesin the previous exercise.