Math 350: Applied Algebra: Codes & Ciphers Spring 2009
Creating BCH (multiple-error-correcting) codes
Theorem 6: Let q(x) be an irreducible polynomial of degree n over GF(p) and let q(x)
also be a primitive polynomial for GF(pn). (i.e. all elements of GF(pn) can be written as
powers of α = root of q(x).) Let α = a primitive element of GF(pn) (so q(α) = 0. ) Let
mi(x) = the minimal polynomials for αi, i = 1, 2, 3, . . . 2t consecutive powers of α.
Let g(x) = lcm{m1(x), m2(x), m3(x), … m2t(x)} be a polynomial of degree r. Let A(x)
={plaintext polynomials of degree ≤ pn – r – 2}.
Then C = the code generated by g(x) = {a(x)*g(x) | a(x) ϵ A(x)} = the BCH code of
designed distance 2t+1. This code will have minimum distance d* ≥ 2t + 1, so C will
correct at least t errors.
q:= x-> x^4 + x + 1;
:= qxx
4
x1
> m3:=(x-a^3)*(x-a^6)*(x-a^9)*(x-a^12);
:= m3 ( )x a
3
( )x a
6
( )x a
9
( )x a
12
> m3:= rem(m3,q(a),a) mod 2;
:= m3 x
4
x
3
x
2
x1
Expand (x*(x+1)*(x^2+x+1)*(x^4+x+1)*(x^4+x^3+1)*(x^4+x^3+x^2+x+1)) mod 2;
x x
16
> Factor (x^64+x) mod 2;
( ) x
6
x
5
x
2
x1 ( )x
6
x1 ( ) x
6
x
4
x
3
x1 ( ) x
6
x
4
x
2
x1
( ) x
6
x
5
1 ( ) x
6
x
5
x
3
x
2
1x( ) x
6
x
5
x
4
x1 ( ) x
3
x
2
1
( ) x
6
x
5
x
4
x
2
1 ( ) x
6
x
3
1 ( )x
3
x1 ( )x
2
x1 ( )x1
q:= x-> x^6 + x + 1:
> m3:=(x-a^3)*(x-a^6)*(x-a^12)*(x-a^24)*(x-a^48)*(x-a^33):
> m3:= rem(m3,q(a),a) mod 2;
:= m3 1x x
4
x
2
x
6
> m5:=(x-a^5)*(x-a^10)*(x-a^20)*(x-a^40)*(x-a^17)*(x-a^34):
> m5:= rem(m5,q(a),a) mod 2;
:= m5 1x
6
x
2
x x
5
> m7:=(x-a^7)*(x-a^14)*(x-a^28)*(x-a^56)*(x-a^49)*(x-a^35):
> m7:= rem(m7,q(a),a) mod 2;
:= m7 1x
6
x
3
> g:=Expand(q(x)*m3*m5*m7) mod 2;
:= g 1x x
2
x
4
x
6
x
5
x
10
x
8
x
13
x
9
x
16
x
17
x
19
x
22
x
20
x
24
x
23
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