1. In this problem we are working over the field Z7, the integers mod 7. All arithmetic is done mod 7. First, write out the addition and multiplication tables for Z7. Secondly, we want codewords to be sequences of elements from Z7. Suppose that in order to detect and correct a possible error we will add two additional digits to the sequence. In this example, we have a sequence of length 3 that we wish to transmit, call it y1, y2, y3, so we find a polynomial fof degree at most 2 for which f(1) = y1, f(2) = y2, f(3) = y3 (all of this is done over Z,.) (To find f we use the 6 functions I discussed in class, but again the arithmetic is done mod 7.) To detect and correct a possible error we compute y4= f (4) and y5 = f(5) and then transmit the sequence y1, y2, y3, y4, y5. If the received sequence is 1, 4, 0, 6, 0 and there is at most one error, determine what the correct sequence is. (Again, remember that you are working mod 7- that will simplify the arithmetic slightly.)

1. In this problem we are working over the field Z7, the integers mod 7. All arithmetic is done mod 7. First, write out the addition and multiplication tables for Z7. Secondly, we want codewords to be sequences of elements from Z7. Suppose that in order to detect and correct a possible error we will add two additional digits to the sequence. In this example, we have a sequence of length 3 that we wish to transmit, call it y1, y2, y3, so we find a polynomial fof degree at most 2 for which f(1) = y1, f(2) = y2, f(3) = y3 (all of this is done over Z,.) (To find f we use the 6 functions I discussed in class, but again the arithmetic is done mod 7.) To detect and correct a possible error we compute y4= f (4) and y5 = f(5) and then transmit the sequence y1, y2, y3, y4, y5. If the received sequence is 1, 4, 0, 6, 0 and there is at most one error, determine what the correct sequence is. (Again, remember that you are working mod 7- that will simplify the arithmetic slightly.)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.5: Congruence Of Integers

Problem 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is...

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