HW8_solns - MATH 3360 Solutions for Selected Porblems in...

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MATH 3360 Solutions for Selected Porblems in Assignment 8 CS 2.12 One can use the sphere-packing bound equation to see if the code is perfect. However, a more intuitive way is this: | C | = 9, d ( C ) = 3 (Check!), and umber of possible words = 81. Since d ( C ) = 2(1)+1, the sphere has radius 1. Therefore, if c = ( a 1 ,a 2 ,a 3 ,a 4 ) C , the sphere containing c contains the words { c, ( a 1 ± 1 ,a 2 ,a 3 ,a 4 ) , ( a 1 ,a 2 ± 1 ,a 3 ,a 4 ) , ( a 1 ,a 2 ,a 3 ± 1 ,a 4 ) , ( a 1 ,a 2 ,a 3 ,a 4 ± 1) } which has 9 elements. So the 9 spheres corresponding to the 9 codes covers 9 × 9 = 81 elements, i.e. the code is perfect. CS 2.19 Note that we want 3 linear dependent columns in the parity check matrix H . No two columns are linearly dependent since (ONLY) in binary code, two vectors are linearly depdendent iff they are equal or one or more of them are zero vectors. 13B 6 i) g ( x ) is a unit is not equivalent to g (1) ,g (2) ,g (3) ,g (0) 6 = 0 , 2. ii) One can easily check (1+2
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HW8_solns - MATH 3360 Solutions for Selected Porblems in...

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