The triple-error-correcting BCH (15, 5) code is constructed on the extended field GF (2 4 ). The extended field GF (2 4 ) is obtained using the primitive polynomial p (x) = x 4 + x 3 + 1. The...

The triple-error-correcting
BCH(15, 5) code is constructed on the extended field
GF(2⁴). The extended field
GF(2⁴) is obtained using the primitive polynomial p (x) = x⁴
+ x³
+ 1. The data-word polynomial is

d(x) . x⁶
+ x³
+ x + 1. Assume that a data-word is encoded and the generated code-word c(x) is transmitted. The received word at the receiver side is expressed in polynomial form as

r(x) = c(x) + e(x) where e(x) = x⁴
+ x². Using r(x), determine the transmitted code-word using the PGZ algorithm.