Question 12: If is a generator polynomial of a BCH code in , prove that divides . [Hint: Exercises 11 and 8(b).]
It has been proved that g(x) divides .
Exercise 11: Let be a finite field of order , whose multiplicative group is generated by . For each i, let be the polynomial of over . If , then each divides .
Exercise 8: If are distinct monic irreducible in such that each divides , then divides
Given that g(x)is a generator polynomial of a BCH code in .
Here, is a product of distinct minimal polynomials .
According to Exercise 11, divides
This implies divides .
Hence, the result.
Complete the proof of Theorem 16.2 by showing that if a code corrects errors, then the Hamming distance between any two codewords is at least . [Hint: If u, v are codewords with , obtain a contradictionby constructing a word w that differs from u in exactly t coordinates and from v in tor fewer coordinates; see Exercise 14.]
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