TIC_2nd_Partial_BCH

162 days ago by avp

x = PolynomialRing(GF(2), 'x').gen() fx = x^6+x^4+x^3+x+1 F = GF(64, 'a', modulus=fx) a=F.gen() 
       
F.is_field() 
        
True
True
fx.is_primitive() 
        
True
True
x = PolynomialRing(GF(2), 'x').gen() F = GF(64, 'a') a = F.gen() 
       
C1 = [mod(2^j, 63) for j in range(6)] print C1 
        
[1, 2, 4, 8, 16, 32]
[1, 2, 4, 8, 16, 32]
C3 = [mod(3*2^j, 63) for j in range(6)] print C3 
        
[3, 6, 12, 24, 48, 33]
[3, 6, 12, 24, 48, 33]
C5 = [mod(5*2^j, 63) for j in range(6)] print C5 
        
[5, 10, 20, 40, 17, 34]
[5, 10, 20, 40, 17, 34]
C7 = [mod(7*2^j, 63) for j in range(6)] print C7 
        
[7, 14, 28, 56, 49, 35]
[7, 14, 28, 56, 49, 35]
C9 = [mod(9*2^j, 63) for j in range(6)] print C9 
        
[9, 18, 36, 9, 18, 36]
[9, 18, 36, 9, 18, 36]
C11 = [mod(11*2^j, 63) for j in range(6)] print C11 
        
[11, 22, 44, 25, 50, 37]
[11, 22, 44, 25, 50, 37]
C13 = [mod(13*2^j, 63) for j in range(6)] print C13 
        
[13, 26, 52, 41, 19, 38]
[13, 26, 52, 41, 19, 38]
C15 = [mod(15*2^j, 63) for j in range(6)] print C15 
        
[15, 30, 60, 57, 51, 39]
[15, 30, 60, 57, 51, 39]
C21 = [mod(21*2^j, 63) for j in range(6)] print C21 
        
[21, 42, 21, 42, 21, 42]
[21, 42, 21, 42, 21, 42]
C23 = [mod(23*2^j, 63) for j in range(6)] print C23 
        
[23, 46, 29, 58, 53, 43]
[23, 46, 29, 58, 53, 43]
C27 = [mod(27*2^j, 63) for j in range(6)] print C27 
        
[27, 54, 45, 27, 54, 45]
[27, 54, 45, 27, 54, 45]
C31 = [mod(31*2^j, 63) for j in range(6)] print C31 
        
[31, 62, 61, 59, 55, 47]
[31, 62, 61, 59, 55, 47]
f1 = prod( x-a^j for j in C1) f3 = prod( x-a^j for j in C3) f5 = prod( x-a^j for j in C5) f7 = prod( x-a^j for j in C7) f9 = prod( x-a^j for j in C9) g = f1*f3*f5*f7*f9 print f1, "\n", f3, "\n", f5, "\n", f7, "\n", f9, "\n", g 
        
x^6 + x^4 + x^3 + x + 1 
x^6 + x^5 + x^4 + x^2 + 1 
x^6 + x + 1 
x^6 + x^3 + 1 
x^6 + x^2 + 1 
x^30 + x^29 + x^27 + x^26 + x^22 + x^21 + x^19 + x^18 + x^16 + x^15 +
x^14 + x^9 + x^8 + x^6 + x^5 + x^2 + 1
x^6 + x^4 + x^3 + x + 1 
x^6 + x^5 + x^4 + x^2 + 1 
x^6 + x + 1 
x^6 + x^3 + 1 
x^6 + x^2 + 1 
x^30 + x^29 + x^27 + x^26 + x^22 + x^21 + x^19 + x^18 + x^16 + x^15 + x^14 + x^9 + x^8 + x^6 + x^5 + x^2 + 1
M = 12*[1]+12*[0]+12*[1] N = sum( M[i]*x^i for i in range(len(M))) print M, "\n", N 
        
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 
x^35 + x^34 + x^33 + x^32 + x^31 + x^30 + x^29 + x^28 + x^27 + x^26 +
x^25 + x^24 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 +
x^2 + x + 1
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 
x^35 + x^34 + x^33 + x^32 + x^31 + x^30 + x^29 + x^28 + x^27 + x^26 + x^25 + x^24 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
def encode(N): return N*x^35 + (N*x^35)%g 
       
encode(N) 
        
x^70 + x^69 + x^68 + x^67 + x^66 + x^65 + x^64 + x^63 + x^62 + x^61 +
x^60 + x^59 + x^46 + x^45 + x^44 + x^43 + x^42 + x^41 + x^40 + x^39 +
x^38 + x^37 + x^36 + x^35 + x^25 + x^24 + x^22 + x^21 + x^20 + x^17 +
x^13 + x^10 + x^9 + x^8 + x^6 + 1
x^70 + x^69 + x^68 + x^67 + x^66 + x^65 + x^64 + x^63 + x^62 + x^61 + x^60 + x^59 + x^46 + x^45 + x^44 + x^43 + x^42 + x^41 + x^40 + x^39 + x^38 + x^37 + x^36 + x^35 + x^25 + x^24 + x^22 + x^21 + x^20 + x^17 + x^13 + x^10 + x^9 + x^8 + x^6 + 1
N1 = x^70 + x^69 + x^68 + x^67 + x^66 + x^65 + x^64 + x^63 + x^62 + x^61 + x^60 + x^59 + x^46 + x^45 + x^44 + x^43 + x^42 + x^41 + x^39 + x^38 + x^37 + x^36 + x^35 + x^25 + x^24 + x^22 + x^21 + x^20 + x^18 + x^17 + x^13 + x^10 + x^9 + x^8 + x^6 + 1 
       
s = sum( N1(a^j)*x^(j-1) for j in [1,2,3,4,5,6,7,8,9] ) print s 
        
x^8 + (a^5 + a^4 + a^2 + a + 1)*x^7 + (a^5 + a^4 + a)*x^6 + (a^5 + a^4 +
a^2 + a + 1)*x^5 + (a^3 + a^2 + 1)*x^4 + a^4*x^3 + a^4*x^2 + a^2*x + a
x^8 + (a^5 + a^4 + a^2 + a + 1)*x^7 + (a^5 + a^4 + a)*x^6 + (a^5 + a^4 + a^2 + a + 1)*x^5 + (a^3 + a^2 + 1)*x^4 + a^4*x^3 + a^4*x^2 + a^2*x + a
def qr(f, g): return (f- f%g)/g, f%g 
       
q1, r1 = qr (x^9,s) print q1, "\n", r1 
        
x + a^5 + a^4 + a^2 + a + 1 
(a^5 + 1)*x^7 + (a^5 + a)*x^6 + (a^4 + a^3 + a^2 + a)*x^5 + (a^5 + a^3 +
a^2 + a + 1)*x^4 + (a^5 + a^4 + a^3 + 1)*x^3 + (a^5 + a^3 + a^2 + 1)*x^2
+ (a^5 + a^4 + a + 1)*x + a^5 + a^4 + a^2 + 1
x + a^5 + a^4 + a^2 + a + 1 
(a^5 + 1)*x^7 + (a^5 + a)*x^6 + (a^4 + a^3 + a^2 + a)*x^5 + (a^5 + a^3 + a^2 + a + 1)*x^4 + (a^5 + a^4 + a^3 + 1)*x^3 + (a^5 + a^3 + a^2 + 1)*x^2 + (a^5 + a^4 + a + 1)*x + a^5 + a^4 + a^2 + 1
q2, r2 = qr (s, r1) print q2, "\n", r2 
        
(a^5 + a^4 + a^3 + a^2 + a + 1)*x + a^3 + a^2 + 1 
a^5 + a^2 + a + 1
(a^5 + a^4 + a^3 + a^2 + a + 1)*x + a^3 + a^2 + 1 
a^5 + a^2 + a + 1
b1 = q1 b2 = 1 + b1*q2 print b2 
        
(a^5 + a^3 + a + 1)*x^9
(a^5 + a^3 + a + 1)*x^9
zeros = [b for b in F if b2(b)==0] print zeros 
        
[a^5 + a^4 + a^3, a^5 + a + 1]
[a^5 + a^4 + a^3, a^5 + a + 1]
error_locators = [discrete_log(b^(-1), a) for b in zeros] print error_locators 
        
[40, 18]
[40, 18]
H = matrix([C1,C3,C5,C7,C9]) print H 
        
[ 1  2  4  8 16 32]
[ 3  6 12 24 48 33]
[ 5 10 20 40 17 34]
[ 7 14 28 56 49 35]
[ 9 18 36  9 18 36]
[ 1  2  4  8 16 32]
[ 3  6 12 24 48 33]
[ 5 10 20 40 17 34]
[ 7 14 28 56 49 35]
[ 9 18 36  9 18 36]
def syndrome(v): return H*vector(v) 
       
P=[0,1,1,1,1,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,1,0,1,1,1,1,0,0] len(P) 
        
63
63
Px = sum( P[i]*x^i for i in range(len(P))) print Px 
        
x^60 + x^59 + x^58 + x^57 + x^55 + x^53 + x^52 + x^51 + x^50 + x^47 +
x^46 + x^41 + x^40 + x^39 + x^35 + x^34 + x^33 + x^29 + x^28 + x^27 +
x^26 + x^24 + x^23 + x^18 + x^11 + x^7 + x^4 + x^3 + x^2 + x
x^60 + x^59 + x^58 + x^57 + x^55 + x^53 + x^52 + x^51 + x^50 + x^47 + x^46 + x^41 + x^40 + x^39 + x^35 + x^34 + x^33 + x^29 + x^28 + x^27 + x^26 + x^24 + x^23 + x^18 + x^11 + x^7 + x^4 + x^3 + x^2 + x
q1, r1 = qr (x^9,s) print q1, "\n", r1 
        
(a^4 + a^2 + a + 1)*x + a^4 + a^2 + 1 
(a^5 + a^3 + a^2 + a + 1)*x^7 + (a + 1)*x^6 + (a^4 + a^3 + a^2 + a +
1)*x^5 + (a^5 + a^3 + a^2 + a)*x^4 + (a^5 + a + 1)*x^3 + (a^5 + a^4 +
a^3 + a^2)*x^2 + a^5*x + a^5 + a^4 + a^2 + 1
(a^4 + a^2 + a + 1)*x + a^4 + a^2 + 1 
(a^5 + a^3 + a^2 + a + 1)*x^7 + (a + 1)*x^6 + (a^4 + a^3 + a^2 + a + 1)*x^5 + (a^5 + a^3 + a^2 + a)*x^4 + (a^5 + a + 1)*x^3 + (a^5 + a^4 + a^3 + a^2)*x^2 + a^5*x + a^5 + a^4 + a^2 + 1
q2, r2 = qr (s, r1) print q2, "\n", r2 
        
(a^5 + a^4)*x + a^3 + a^2 + a 
(a^3 + a^2)*x^6 + (a^5 + a^2 + a)*x^5 + (a^3 + a + 1)*x^4 + (a^5 + a^2 +
a)*x^3 + (a^4 + a^3 + a^2 + a + 1)*x^2 + (a + 1)*x + a^4 + a + 1
(a^5 + a^4)*x + a^3 + a^2 + a 
(a^3 + a^2)*x^6 + (a^5 + a^2 + a)*x^5 + (a^3 + a + 1)*x^4 + (a^5 + a^2 + a)*x^3 + (a^4 + a^3 + a^2 + a + 1)*x^2 + (a + 1)*x + a^4 + a + 1
q3, r3 = qr (r1,r2) print q3, "\n", r3 
        
(a^3 + a + 1)*x + a^4 + a 
(a^4 + a^3)*x^5 + a^3*x^4 + (a^3 + a^2 + a)*x^3 + (a^5 + a^4 + a^2 +
1)*x^2 + (a^3 + a^2 + 1)*x + a^4 + a^2
(a^3 + a + 1)*x + a^4 + a 
(a^4 + a^3)*x^5 + a^3*x^4 + (a^3 + a^2 + a)*x^3 + (a^5 + a^4 + a^2 + 1)*x^2 + (a^3 + a^2 + 1)*x + a^4 + a^2
q4, r4 = qr (r2,r3) print q4, "\n", r4 
        
(a^5 + a^3 + a^2 + 1)*x + a^4 + a^2 + 1 
(a^4 + a)*x^4 + (a^5 + a^3 + a)*x^3 + (a^5 + a^4 + 1)*x^2 + (a^3 + a^2 +
1)*x + a^5
(a^5 + a^3 + a^2 + 1)*x + a^4 + a^2 + 1 
(a^4 + a)*x^4 + (a^5 + a^3 + a)*x^3 + (a^5 + a^4 + 1)*x^2 + (a^3 + a^2 + 1)*x + a^5
q5, r5 = qr (r3,r4) print q5, "\n", r5 
        
(a^5 + a + 1)*x + a^4 + a^3 + a^2 + 1 
(a^3 + 1)*x^3 + (a^5 + a^4)*x
(a^5 + a + 1)*x + a^4 + a^3 + a^2 + 1 
(a^3 + 1)*x^3 + (a^5 + a^4)*x
q6, r6 = qr (r4,r5) print q6, "\n", r6 
        
a*x + a^5 + a^4 + a + 1 
(a^3 + a)*x^2 + a^5
a*x + a^5 + a^4 + a + 1 
(a^3 + a)*x^2 + a^5
q7, r7 = qr (r5,r6) print q7, "\n", r7 
        
(a^4 + a^3 + a)*x 
(a^5 + a^3 + 1)*x
(a^4 + a^3 + a)*x 
(a^5 + a^3 + 1)*x
q8, r8 = qr (r6,r7) print q8, "\n", r8 
        
(a^5 + a^4 + a^3 + a^2 + a)*x 
a^5
(a^5 + a^4 + a^3 + a^2 + a)*x 
a^5
q9, r9 = qr (r7,r8) print q9, "\n", r9 
        
(a^5 + a^4 + a^2 + a)*x 
0
(a^5 + a^4 + a^2 + a)*x 
0
b1 = q1 b2 = 1 + b1*q2 b3 = b1 + b2*q3 b4 = b2 + b3*q4 b5 = b3 + b4*q5 b6 = b4 + b5*q6 b7 = b5 + b6*q7 b8 = b6 + b7*q8 b9 = b7 + b8*q9 print b4 
        
a^4*x^4 + (a^5 + a^4 + a^3 + 1)*x^3 + (a^4 + a)*x^2 + (a^5 + a + 1)*x +
a^5 + a
a^4*x^4 + (a^5 + a^4 + a^3 + 1)*x^3 + (a^4 + a)*x^2 + (a^5 + a + 1)*x + a^5 + a
zeros = [b for b in F if b4(b)==0] print zeros 
        
[a^2]
[a^2]
error_locators = [discrete_log(b^(-1), a) for b in zeros] print error_locators 
        
[61]
[61]