ps7 calcs.ipynb

Kernel: Python 3 (ipykernel)

k.<z2>=GF(4)
R.<T> = PolynomialRing(k)
f = T^11 - 1

l.<w>  = GF(4^5)
n = int((4^5-1)/11)

z= w^n

multiplicative_order(z)

11

#product([ T-z^(4^i) for i in range(5) ])

f.factor()

(T + 1) * (T^5 + z2*T^4 + T^3 + T^2 + (z2 + 1)*T + 1) * (T^5 + (z2 + 1)*T^4 + T^3 + T^2 + z2*T + 1)

fs = list(f.factor())
fs

[(T + 1, 1),
 (T^5 + z2*T^4 + T^3 + T^2 + (z2 + 1)*T + 1, 1),
 (T^5 + (z2 + 1)*T^4 + T^3 + T^2 + z2*T + 1, 1)]

(g,_) = fs[1]
(h,_) = fs[2]

(g,h)

(T^5 + z2*T^4 + T^3 + T^2 + (z2 + 1)*T + 1,
 T^5 + (z2 + 1)*T^4 + T^3 + T^2 + z2*T + 1)

S.<X>  = PolynomialRing(l)
gg=product([ X-z^(4^i) for i in range(5) ])

hh = product([X- z^(2*4^i) for i in range(5)])

(gg,hh)

(X^5 + (w^5 + w^3 + w)*X^4 + X^3 + X^2 + (w^5 + w^3 + w + 1)*X + 1,
 X^5 + (w^5 + w^3 + w + 1)*X^4 + X^3 + X^2 + (w^5 + w^3 + w)*X + 1)

(4^5-1)/3

341

m = X^2 + X + 1
m(w^5 + w^3 + w)

0

V = VectorSpace(k,11)

def pad(ll,n):
    # pad the list ll with 0's to make it have length n
    x = len(ll)
    if x < n:
        return ll + (n-x)*[0]
    else:
        return ll[0:n]

def vectorize(p,n):
    # make a vector of length n out of a polynomial
    coeffs = p.coefficients(sparse=False)
    return V(pad(coeffs,n))


def mkCode(p):
    # vectorize the polynomial T^i * p and use the  vectors as a basis for the code C
    # I'm assuming deg p = 5...
    return V.subspace([ vectorize( T^i * p, 11) for i in range(6) ])

C1 = mkCode(g)
C2 = mkCode(h)

def weight(v):
    r = [x for x in v if x != 0]
    return len(r)

def min_distance(D):
    # brute-force computation of minimal distance of D
    return min([ weight(v) for v in D if v != 0])

[ min_distance(c) for c in [C1,C2]]

[5, 5]