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 f of degree at most 2 for which f(1)=y1,f(2)=y2,f(3)=y3 (all of this is done over Z7.) (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.)

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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 f of degree at most 2 for which f(1)=y1,f(2)=y2,f(3)=y3 (all of this is done over Z7.) (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.)

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Consider the field Z7={0,1,2,3,4,5,6}mod7Solution:Consider the field Z7={0,1,2,3,4,5,6}mod7We first write the addition and multiplication tables for Z7+012345600123456112345602234560133456012445601235560123466012345The multiplicative Tables for Z7×012345600000000101234562024613530362514404152635053164260654321

In this problem we are working over the field Z_{7}, the integers mod 7. All arithmetic is don

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