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EE4212~tut1_sol

# EE4212~tut1_sol - q-1=4 They are primitive elements and can...

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1 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 2 ] [ dt bt bt at + = 1 0 2 2 ] ) [( dt bt t b a + + + = 1 0 2 2 3 4 2 ] ) ( 2 ) [( dt t b t b a b t b a 1 0 3 2 1 0 4 1 0 5 2 3 2 ) ( 5 ) ( t b t b a b t b a + + + = 3 2 ) ( 5 ) ( 2 2 b b a b b a + + + = Example : a = 1, b = 4 => shaded area = 3 1

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2 2(a) Order of each element in a finite field GF(5) => q =5 with elements 0, 1, 2, 3, 4 1 1 = 1 = 1 mod 5 Order of the element 1 is 1 2 4 = 16 = 1 mod 5 Order of the element 2 is 4 3 4 = 81 = 1 mod 5 Order of the element 3 is 4 4 2 = 16 = 1 mod 5 Order of the element 4 is 2 q =5 => q -1=4 => order of an element should divide 4 => can be 1, 2 or 4 only 2(b) Primitive elements of a finite field GF(5) : elements 2 and 3 have order
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Unformatted text preview: q-1=4. They are primitive elements and can generate all the non-zero elements in GF(5). 2 1 = 2 = 2 mod 5 3 1 = 3 = 3 mod 5 2 2 = 4 = 4 mod 5 3 2 = 9 = 4 mod 5 2 3 = 8 = 3 mod 5 3 3 = 27 = 2 mod 5 2 4 = 16 = 1 mod 5 3 4 = 81 = 1 mod 5 2(c) Multiplicative inverse of each element 1 * 1 = 1 = 1 mod 5 Multiplicative inverse of the element 1 is itself 2 * 3 = 6 = 1 mod 5 Multiplicative inverse of the element 2 is 3 3 * 2 = 6 = 1 mod 5 Multiplicative inverse of the element 3 is 2 4 * 4 = 16 = 1 mod 5 Multiplicative inverse of the element 4 is itself...
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EE4212~tut1_sol - q-1=4 They are primitive elements and can...

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