Example 1 GF(q) Random Code.sagews

#
# To test the encoding and decoding processes of GF(q) Random code.
#
# by Chong Zan Kai
# 25-Oct-2015
#


#-------------------------------------------------------------------------------
# Configuration
#-------------------------------------------------------------------------------
q = 2**1
k = 100 # total message symbols
l = 100 # symbol size
GFq = GF(q, 'x')
extra_rows = 10 # Generate how many extra rows.


def generate_encoded_symbol(message_mat):
    '''
    Generate one encoded symbol.
    Return: (generator vector, encoded vector)
    '''
    total_row = message_mat.nrows()
    generator_vec = random_vector(GFq, total_row)

    encoded_vec = generator_vec * message_mat
    return (generator_vec, encoded_vec)


def GFq_solve_matrix(generator_mat, data_mat = None, duplicate = False):
    '''
    Inverse the GFq random matrix with Gaussian elimination.
        generator_mat: generator matrix.
        data_mat: the data matrix that associated with generator matrix.
        duplicate: Work on the copy of generator_mat and data_mat if set to true.
    Return
    '''
    # Duplication, if required.
    if duplicate == True:
        generator_mat = copy(generator_mat)
        data_mat = copy(data_mat)

    # Forming the triangular matrix.
    for col in range(generator_mat.ncols()):
        # Building pivoting row, e.g. [0,1,0,0...]
        if generator_mat[col][col] ==  0:

            # Need to ensure at least the "leading nonzero" is there.
            # If the row has no "leading non-zero", get a new row that has the "leading non-zero".
            nonzero_row = -1
            for i in range(col, generator_mat.nrows()):
                if generator_mat[i][col] != 0: # found it.
                    nonzero_row = i
                    break
            if nonzero_row == -1: # Fail. Nothing can do.
                return None, None

            # Swap rows in between Row(col) and Row(nonzero_row).
            generator_mat.swap_rows(col, nonzero_row)

            # Process the data matrix if provided.
            if data_mat:
                data_mat.swap_rows(col, nonzero_row)
        # End if

        # Make sure  the pivoting row with leading 1.
        if generator_mat[col][col] != 0:
            multiplier = 1 / generator_mat[col][col]
            generator_mat[col] *= multiplier
            # Process the data matrix if provided.
            if data_mat:
                data_mat[col] *= multiplier

        # Clear the column of the leading 1.
        for j in [x for x in range(generator_mat.nrows()) if x != col]:
            if generator_mat[j][col] != 0:
                multiplier = generator_mat[j][col]
                generator_mat[j] = generator_mat[j] + multiplier * generator_mat[col]
                # Process the data matrix if provided.
                if data_mat:
                    data_mat[j] = data_mat[j] + multiplier * data_mat[col]

    # End for loop.
    return generator_mat, data_mat


# def backward_substitution(generator_mat, data_mat = None):
#     # Starting from the rightmost column.
#     for col_num in reversed(range(1, generator_mat.ncols())):
# #         print 'col_num = %d' % col_num
#         # The row numbers should start from (col_num-1) to 0.
#         for row_num in reversed(range(col_num)):
# #             print 'row_num = %d' % (row_num)
#             # Clean all the rows above, in which thir $col_num is nonzero.
#             if generator_mat[row_num][col_num] != 0:
#                 multiplier = generator_mat[row_num][col_num]
#                 generator_mat[row_num] = generator_mat[row_num] + (multiplier ) * generator_mat[col_num]
#                 if data_mat:
#                     data_mat[row_num] = data_mat[row_num] + (multiplier ) * data_mat[col_num]
#     return generator_mat, data_mat

#-------------------------------------------------------------------------------
# Preparing a message
#-------------------------------------------------------------------------------

message_mat = random_matrix(GFq, k,l)

print 'message_mat =\n%s' % message_mat.str()
print

#-------------------------------------------------------------------------------
# Generate sufficient (i.e. $extra_row) encoded symbols.
#-------------------------------------------------------------------------------


generator_vec_list = []
encoded_vec_list = []
for i in (1 .. message_mat.nrows() + extra_rows):
    generator_vec, encoded_vec = generate_encoded_symbol(message_mat)
    generator_vec_list.append(generator_vec)
    encoded_vec_list.append(encoded_vec)

encoded_mat = matrix(encoded_vec_list)
generator_mat = matrix(generator_vec_list)

print 'generator_mat = \n%s' % generator_mat.str()
print

print 'encoded_mat = \n%s' % encoded_mat.str()
print


#-------------------------------------------------------------------------------
# Decoding the message
#-------------------------------------------------------------------------------
g_mat, d_mat = GFq_solve_matrix(generator_mat, encoded_mat)

assert (g_mat != None)
assert (d_mat != None)

#-------------------------------------------------------------------------------
# Compare whether the decoded message is correct.
#-------------------------------------------------------------------------------
decoded_mat = d_mat.matrix_from_rows(range(message_mat.nrows()))
print 'decoded_mat = \n%s' % decoded_mat.str()
print

print 'message_mat = \n%s' % message_mat.str()
print

print 'Comparing the decoded_mat and message_mat:',
if decoded_mat == message_mat:
    print 'Decoding successfully.'
else:
    print 'Decoding fail.'

message_mat = [1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 0 1 1 0 1 0] [1 1 1 1 1 1 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1] [1 0 1 1 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 1 1 1 1 1 0 1] [1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 1 1 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 1 0 0 0 0 0 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1 1 0 0 1 1 1 0 0 1 0 0 0 1 0 1 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0] Comparing the decoded_mat and message_mat: Decoding successfully.