cyclicCodes.sagews

%typeset_mode True

k=IntegerModRing(2)

k(1)+k(1)

$\displaystyle 0$

R.<x>=k[]

(x+1)*(x^3+x+1)*(x^3+x^2+1)*(x^4+x^3+x^2+x+1)

$\displaystyle x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{4} + x^{3} + x^{2} + x + 1$

(x^3+x^2+1)*(x^4+x^3+x^2+x+1)

$\displaystyle x^{7} + x^{4} + x^{3} + x + 1$

factor(x^8+x^7+1)

$\displaystyle (x^{2} + x + 1) \cdot (x^{6} + x^{4} + x^{3} + x + 1)$

G=matrix(k,
         [[1,1,1,0,0,0,0,0,0,0,0],
          [0,1,1,1,0,0,0,0,0,0,0],
          [0,0,1,1,1,0,0,0,0,0,0],
          [0,0,0,1,1,1,0,0,0,0,0],
          [0,0,0,0,1,1,1,0,0,0,0],
          [0,0,0,0,0,1,1,1,0,0,0],
          [0,0,0,0,0,0,1,1,1,0,0],
          [0,0,0,0,0,0,0,1,1,1,0],
          [0,0,0,0,0,0,0,0,1,1,1]])
H=transpose(matrix(k,
         [[1,0,1,1,0,1,1,0,1,1,0],
          [1,1,0,1,1,0,1,1,0,1,1]]))
G
H

$\displaystyle \left(\begin{array}{rrrrrrrrrrr} 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \end{array}\right)$

$\displaystyle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \\ 0 & 1 \end{array}\right)$

(G*H)

$\displaystyle \left(\begin{array}{rr} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right)$

C=LinearCode(G);

C

$\displaystyle [11, 9]\textnormal{ Linear code over }\ZZ/2\ZZ$

C.minimum_distance()

$\displaystyle 2$

G2=transpose(H)
C2=LinearCode(G2)
C2

$\displaystyle [11, 2]\textnormal{ Linear code over }\ZZ/2\ZZ$

C2.minimum_distance()

$\displaystyle 7$

factor(x^11+1)

$\displaystyle {\left(x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\right)} {\left(x + 1\right)}$

S = R.quotient(x^31+1, 'a')
a = S.gen()

g=(a+1)*(a^5+a^4+a^3+a^2+1)

h = g^31
h

$\displaystyle a^{26} + a^{22} + a^{21} + a^{20} + a^{18} + a^{16} + a^{13} + a^{11} + a^{10} + a^{9} + a^{8} + a^{5} + a^{4} + a^{2} + a + 1$

w = (x^26+x^22+x^21+x^20+x^18+x^16+x^13+x^11+x^10+x^9+x^8+x^5+x^4+x^2+x+1)
w

$\displaystyle x^{26} + x^{22} + x^{21} + x^{20} + x^{18} + x^{16} + x^{13} + x^{11} + x^{10} + x^{9} + x^{8} + x^{5} + x^{4} + x^{2} + x + 1$

factor(w)

$\displaystyle (x + 1)^{3} \cdot (x^{2} + x + 1)^{2} \cdot (x^{5} + x^{4} + x^{3} + x^{2} + 1) \cdot (x^{7} + x^{4} + x^{3} + x^{2} + 1)^{2}$

h^2

$\displaystyle a^{26} + a^{22} + a^{21} + a^{20} + a^{18} + a^{16} + a^{13} + a^{11} + a^{10} + a^{9} + a^{8} + a^{5} + a^{4} + a^{2} + a + 1$