tut1_sol - generate all the non-zero elements in GF(7 3 1 =...

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EE 4212 Information and Coding Solution to Tutorial 1 1. Discrete form of the Euclidian metric: = k k k y x y x d 2 ) ( ) , ( Generalized it to continuous case: = 1 0 2 )] ( ) ( [ )] ( ), ( [ dt t y t x t y t x d = 1 0 2 2 ] ) ( [ dt t a t b + + = 1 0 2 2 2 ] ) 2 1 ( [ dt b a t ab bt + + + + + + = 1 0 2 4 2 2 2 2 2 3 4 2 ] ) 2 1 ( 2 ) ) 2 1 ( 2 ( ) 2 1 ( 2 [ dt b a t ab b a t ab b a t ab b t b 1 0 2 4 1 0 2 2 1 0 3 2 2 1 0 4 1 0 5 2 2 ) 2 1 ( 2 3 ) 6 4 1 ( 4 ) 2 1 ( 2 5 t b a t ab b a t b a ab t ab b t b + + + + + + = 2 4 2 2 2 2 ) 2 1 ( 3 ) 6 4 1 ( 2 ) 2 1 ( 5 b a ab b a b a ab ab b b + + + + + + = Example : a = 0.5, b = 4 => shaded area = 0.4472
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2(a) Order of each element in a finite field GF(7) => q =7 with elements 0, 1, 2, 3, 4, 5, 6 1 1 = 1 = 1 mod 7 Order of the element 1 is 1 2 3 = 8 = 1 mod 7 Order of the element 2 is 3 3 6 = 729 = 1 mod 7 Order of the element 3 is 6 4 3 = 64 = 1 mod 7 Order of the element 4 is 3 5 6 = 15625 = 1 mod 7 Order of the element 5 is 6 6 2 = 36 = 1 mod 7 Order of the element 6 is 2 q =7 => q -1=6 => order of an element should divide 6 => can be 1, 2, 3 or 6 2(b) Primitive elements of a finite field GF(7) : elements 3 and 5 have order q -1=6. They are primitive elements and can
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Unformatted text preview: generate all the non-zero elements in GF(7). 3 1 = 3 = 3 mod 7 5 1 = 5 = 5 mod 7 3 2 = 9 = 2 mod 7 5 2 = 25 = 4 mod 7 3 3 = 27 = 6 mod 7 5 3 = 125 = 6 mod 7 3 4 = 81 = 4 mod 7 5 4 = 625 = 2 mod 7 3 5 = 243 = 5 mod 7 5 5 = 3125 = 3 mod 7 3 6 = 729 = 1 mod 7 5 6 = 15625 = 1 mod 7 2(c) Multiplicative inverse of each element 1 * 1 = 1 = 1 mod 7 Multiplicative inverse of the element 1 is 1 2 * 4 = 8 = 1 mod 7 Multiplicative inverse of the element 2 is 4 3 * 5 = 15 = 1 mod 7 Multiplicative inverse of the element 3 is 5 4 * 2 = 8 = 1 mod 7 Multiplicative inverse of the element 4 is 2 5 * 3 = 15 = 1 mod 7 Multiplicative inverse of the element 5 is 3 6 * 6 = 36 = 1 mod 7 Multiplicative inverse of the element 6 is itself...
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