PROBLEMS 6.1 Consider the Galois field GF(24) given by Table 2.8. The element 8 = a is also a primitive element. Let go(X) be the lowest-degree polynomial over GF(2) that has B.B2.83.84 as its roots. This polynomial also generates a double-error-correcting primitive BCH code of length 15. a. Determine g(X). b. Find the parity-check matrix for this code. c. Show that go(X) is the reciprocal polynomial of the polynomial g(X) that generates the (15, 7) double-error-correcting BCH code given in Example 6.1. TABLE 2.8: Three representations for the elements of GF(24) generated by p(X) = 1+X+X4. Power representation Polynomial representation α 1 0 0 1 4-Tuple representation (0000) α (1000) a² (0100) Q3 (0010) Q4 α3 1 + α (0001) аб a+a² (1100) α7 a² + a³ (0110) 1+α (0011) Q8 1 Q9 +α² (1101) (1010) α α10 1+a+a² (0101) a11 (1110) α12 α13 1+a+a²+a³ (0111) 1 (1111) 14 1 (1011) (1001) PROBLEMS 6.1 Consider the Galois field GF(24) given by Table 2.8. The element ẞ = a is also a primitive element. Let go(X) be the lowest-degree polynomial over GF(2) that has 8.82.83.84 as its roots. This polynomial also generates a double-error-correcting primitive BCH code of length 15. a. Determine go(X). b. Find the parity-check matrix for this code. c. Show that g(X) is the reciprocal polynomial of the polynomial g(X) that generates the (15, 7) double-error-correcting BCH code given in Example 6.1.