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Math 342 Homework Assignment #5 (Due Thursday, March 24) 1. For each linear code C, over GF (q), defined by the following generator matrices, find a parity check matrix for C and generator and parity check matrices for C ⊥ . (a) q = 3 1 0 0 0 1 2 G= 0 1 0 2 0 2 0 0 1 2 0 0 (b) q = 4 0 1 0 1 a b G= b 0 0 1 1 0 a a a a b 1 2. Find the minimum distance of the linear codes over GF (q) defined by each of the following parity check matrices. (a) q = 3 H= 2 2 2 1 1 0 2 2 (b) q = 4 a b 1 1 H= b b a 0 1 a a b 3. Let C be the linear code with parity check matrix 1 1 1 1 0 0 H= 0 1 1 0 1 0 1 1 0 0 0 1 (a) Construct a syndrome decoding table, i.e., for each coset of C find a coset leader and the corresponding syndrome. (b) Decode each of the following received vectors i. 110010 ii. 111111 4. Let C and D be linear codes in V (n, q). Let C + D = {x + y : x ∈ C, y ∈ D}. (a) Show that C + D is a linear code. (b) Show that (C + D)⊥ = C ⊥ ∩ D⊥ (c) Show that (C ∩ D)⊥ = C ⊥ + D⊥ 1