research: algebraic codes

script/research/codes/decode-berkle.sage

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+import itertools
+
+q = 11
+n = 10
+k = 4
+d = n - k + 1
+
+K = GF(q)
+R.<x> = K[]
+α = K(2)
+assert set(α^i for i in range(q - 1)) | {0} == set(K)
+
+# Pick a random codeword
+m = (3, 0, 7, 9)
+f = m[0] + m[1]*x + m[2]*x^2 + m[3]*x^3
+print(f"m = {m}")
+
+c = vector(f(α^i) for i in range(q - 1))
+assert len(c) == q - 1
+print(f"c = {c}")
+
+e = int((n - k)/2)
+assert e == 3
+# We can tolerate <= (n-k)/2 = 3 errors
+c[2] = 0
+c[4] = 6
+c[7] = 7
+
+# Naive and very slow
+freqs = {}
+for (i0, i1, i2, i3) in itertools.permutations(range(n), int(4)):
+    g = R.lagrange_polynomial([
+        (α^i0, c[i0]),
+        (α^i1, c[i1]),
+        (α^i2, c[i2]),
+        (α^i3, c[i3]),
+    ])
+    if not g in freqs:
+        freqs[g] = 0
+    freqs[g] += 1
+max_key = max(freqs.keys(), key=lambda k: freqs[k])
+assert max_key == f
+
+E0 = (x - α^2)*(x - α^4)*(x - α^7)
+assert E0.degree() <= n - k - 1
+N0 = E0*f
+print(f"E = {E0}")
+print(f"N = {N0}")
+
+S = []
+for i in range(n):
+    row = []
+
+    αi = α^i
+
+    # deg N = e + (k - 1)
+    for j in range(e + k):
+        row.append(αi^j)
+
+    r_i = c[i]
+    # deg E = e
+    # We don't need x^e here
+    for j in range(e):
+        row.append(-r_i * αi^j)
+
+    assert n == 2*e + k
+    assert len(row) == n
+    S.append(row)
+
+assert len(S) == n
+A = matrix(S)
+s = vector(r * α^(i*e) for (i, r) in enumerate(c))
+print(f"s = {s}")
+v = A.solve_right(s)
+print(f"A = {A}")
+print(f"v = {v}")
+
+Nv, Ev = v[:e + k], v[e + k:]
+print(Nv)
+print(Ev)
+N = sum(Ni * x^i for (i, Ni) in enumerate(Nv))
+E = sum(Ei * x^i for (i, Ei) in enumerate(Ev)) + x^e
+print(f"N = {N}")
+print(f"E = {E}")
+assert N == N0
+assert E == E0
+
+f, rem = N.quo_rem(E)
+assert rem == 0
+print(f"f = {f} =", list(f))
+

script/research/codes/decode-simple.sage

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+import itertools
+from tabulate import tabulate
+
+q = 5
+n = 4
+k = 2
+d = n - k + 1
+
+K = GF(q)
+R.<x> = K[]
+α = K(2)
+assert set(α^i for i in range(q - 1)) | {0} == set(K)
+
+# Pick a random codeword
+m = (3, 2)
+f = m[0] + m[1]*x
+
+c = [f(α^i) for i in range(n)]
+assert len(c) == n
+
+# We can tolerate <= (n-k)/2 = 1 error
+c[1] = 0
+
+# f is degree 3 so we need 4 points to reconstruct it
+table = []
+count = 0
+total = 0
+for (i0, i1) in itertools.permutations(range(n), int(2)):
+    g = R.lagrange_polynomial([(α^i0, c[i0]), (α^i1, c[i1])])
+    table.append([
+        (i0, i1),
+        g,
+        "*" if f == g else None
+    ])
+    if f == g:
+        count += 1
+    total += 1
+print(tabulate(table))
+print(f"{count} / {total}")
+

script/research/codes/punch.sage

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+q = 11
+k = 3
+d0 = q - k + 1
+
+s = 4
+assert s <= d0 - 1
+n = q - s
+d = n - k + 1
+
+K = GF(q)
+F.<z> = K[]
+
+V = VectorSpace(K, n)
+C = V.subspace([
+    [1, 0, 0, 0, 0, 0, 0],
+    [0, 1, 0, 0, 0, 0, 0],
+    [0, 0, 1, 0, 0, 0, 0],
+])
+for c in C:
+    f = c[0] + c[1]*z + c[2]*z^2
+
+    w = vector(f(β) for β in list(K)[:n])
+    assert len(w) == n
+
+    if w.is_zero():
+        continue
+
+    assert d <= w.hamming_weight()
+