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practicemidterm312 - G denote a group having the just the 4...

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PRACTICE MIDTERM FOR MAT 312 (1) Let C Z 20 2 denote a set of code words. Suppose that d = 7 for this code. (a) How many transmission errors can be detected by this code? Explain why. (b) How many transmission errors can be corrected by this code? Ex- plain why. (c) Give a simple example of a code C with d = 7. (2) Let H denote a 4 × 6 binary matrix, and let C Z 6 2 denote all the binary 6-tuples c such that H c t = 0 . (a) Explain why C is a group code. In (b)-(e) below assume that H is equal to 0 0 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 1 1 1 1 Try do (b)-(d) without listing the code words in C . (b) Show that C is single error detecting and single error correcting. (c) Compute d for this code. (d) A code word c is transmitted and a binary 6-tuple r is received. If r = 110000 then compute the syndrome of r ; if at most one trans- mission error has been made, then find c . (e) List all the code words in C . (3)
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Unformatted text preview: G denote a group having the just the 4 elements G = { a, b, c, d } . Suppose that ab = a and a 3 = c . (a) Which element of G is the identity. (b) ±ill in the multiplication table for G . (c) Is G isomorphic to Z 4 ? Is G isomorphic to Z 2 × Z 2 ? (4) Explain why each of the following pairs of groups are (or are not) iso-morphic. (a) Z 6 and Z 2 × Z 3 . (b) S 3 and Z 6 . (c) S 4 and S 3 . (d) S 2 and Z 2 . 1 2 PRACTICE MIDTERM FOR MAT 312 (5) Consider the permutation σ ∈ S 8 defned by σ = (126)(8241)(5368). (a) Write σ as a 2 × 8 matix having 1 2 3 4 5 6 7 8 For frst row. (b) Write σ-1 as a 2 × 8 matrix (as in part (a)). (c) Compute order ( σ ). (d) Write σ as a product oF disjoint cycles....
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