Abstract Algebra Theory and Applications - Computer Science ...

134 CHAPTER 7 ALGEBRAIC CODING THEORY13. Let⎛H = ⎝0 1 1 1 10 0 0 1 11 0 1 0 1Compute the syndrome caused by each of the following transmission errors.(a) An error in the first bit(b) An error in the third bit(c) An error in the last bit(d) Errors in the third and fourth bits14. Let C be the group code in Z 3 2 defined by the codewords (000) and (111).Compute the cosets of H in Z 3 2. Why was there no need to specify right orleft cosets? Give the single transmission error, if any, to which each cosetcorresponds.15. For each of the following matrices, find the cosets of the corresponding codeC. Give a decoding table for each code if possible.⎛(a)⎝ 0 1 0 0 0⎞⎛⎞(b) 0 0 1 0 01 0 1 0 1 ⎠⎜ 1 1 0 1 0⎟1 0 0 1 0⎝ 0 1 0 1 0 ⎠1 1 0 0 1⎞⎠ .(c)( 1 0 0 1 10 1 0 1 1)(d)⎛⎜⎝1 0 0 1 1 1 11 1 1 0 0 1 11 0 1 0 1 0 11 1 1 0 0 1 0⎞⎟⎠16. Let x, y, and z be binary n-tuples. Prove each of the following statements.(a) w(x) = d(x, 0)(b) d(x, y) = d(x + z, y + z)(c) d(x, y) = w(x − y)17. A metric on a set X is a map d : X × X → R satisfying the followingconditions.(a) d(x, y) ≥ 0 for all x, y ∈ X;(b) d(x, y) = 0 exactly when x = y;(c) d(x, y) = d(y, x);(d) d(x, y) ≤ d(x, z) + d(z, y).