using code III, decode the received word R = (1,1,0,1,1,0,0). we form the polyno

Question

using code III, decode the received word R = (1,1,0,1,1,0,0).
we form the polynomial R(x) =1*x^6 + 1*x^5 + 0*x^4 + 1*x^3 + 1*x^2 + 0*x + 0 = x^6+x^5+x^3+x^2

we find:
R(a) a^6 + a^5 +a^3 +a^2 = (a^2 + a)+(a^2+a+1)+(a+1)+a^2 = a^2 + a = a^4

since R(a) a^4, we change the coefficient of x^4 in R(x) and conclude that C(x) was C(x)=1*x^6 + 1*x^5 + 1*x^4 + 1*x^3 + 1*x^2 + 0*x + 0 = x^6 + x^5 + x^4 + x^3 + x^2.

we conclude that the orginal message word is (1, 1, 1, 1) the coefficients of the leftmost foursr powers of x in C(x). [this is correct if at most one digit was changed in the transmission of C]

Explanation / Answer

R(x)= x^6+x^5+x^3+x^2

and we decode it as R(x) = x^4

C(x) = x^6+x^5+x^4+x^3+x^2

C(x) = (x^2+x+1)+(x+1)+x^2

C(x) = x^2+x

C(x) = x^4

so if we decode the C(x) we get the answer as x^4

after decoding the value of C(x) = x^4

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