4. (10 pts)
BCH Code Design.
The following table lists the conjugates and their corresponding minimal polynomials
for GF(64).
Table 1: Cyclotomic cosets (conjugates) and their minimal polynomials for GF(64).
Conjugacy Class
Minimal Polynomial
1
x
+ 1
α, α
2
, α
4
, α
8
, α
16
, α
32
x
6
+
x
+ 1
α
3
, α
6
, α
12
, α
24
, α
33
, α
48
x
6
+
x
4
+
x
2
+
x
+ 1
α
5
, α
10
, α
17
, α
20
, α
34
, α
40
x
6
+
x
5
+
x
2
+
x
+ 1
α
7
, α
14
, α
28
, α
35
, α
49
, α
56
x
6
+
x
3
+ 1
α
9
, α
18
, α
36
x
3
+
x
2
+ 1
α
11
, α
22
, α
25
, α
37
, α
44
, α
50
x
6
+
x
5
+
x
3
+
x
2
+ 1
α
13
, α
19
, α
26
, α
38
, α
41
, α
52
x
6
+
x
4
+
x
3
+
x
+ 1
α
15
, α
30
, α
39
, α
51
, α
57
, α
60
x
6
+
x
5
+
x
4
+
x
2
+ 1
α
21
, α
42
x
2
+
x
+ 1
α
23
, α
29
, α
43
, α
46
, α
53
, α
58
x
6
+
x
5
+
x
4
+
x
+ 1
α
27
, α
45
, α
54
x
3
+
x
+ 1
α
31
, α
47
, α
55
, α
59
, α
61
, α
62
x
6
+
x
5
+ 1
Design a binary BCH code with length
n
= 63 for design distance
δ
= 8.
You may
express the generator polynomial as a product of minimal polynomials.
To get full
credit you should find the highest rate code possible.