CoCalc -- Test Eq 3.sagews

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#
# Examine the Eq 3 in “Performance versus Overhead for Fountain Codes over Fq”  Liva et al. (2010)
# That is the equation for P_error of Random Linear Fountain (RLF) Code.
#
# by Chong ZK
# 24-Oct-2015
#
# Public link: https://cloud.sagemath.com/projects/296436a6-28bf-4a04-986b-727ebc30167f/files/Test%20Eq%203.sagews
#
def liva_p_err(k, delta, q):
    '''
    Equation 3 of “Performance versus Overhead for Fountain Codes over Fq”  Liva et al. (2010).
    Measuring the P_error of RLF code.
    k : total message symbols.
    delta : total overhead symbols.
    q : GF(q).
    '''
    result = 1.0 - prod( 1 - (q**(i-1)) / q**(k+delta) for i in (1 .. k) )
    return result

def zk_p_err(k, m, q):
    '''
    To calculate the failure probability to have rank k in a GF(q) random matrix of dimensions of (k+m) x k.
    '''
    val = 1.0 - prod((1.0 - q**(i-k)) for i in (1 .. k-m-1))
    return val

#--------------------------------------------------------------------
# Testing
#--------------------------------------------------------------------
#
# I would like to the P_success and P_error of RLF code for the first ten overhead symbols.
#

k = 10 # Total message symbols
q = 2**2  # GF(q)

print '# Live''s equation'
# delta is the total overhead symbols
for delta in (0 .. 10):
    p_error = liva_p_err(k, delta, q)
    p_success = 1 - p_error
    print 'In GF(%d), k=%d, delta=%d, p_success=%f, p_error=%f' % (q, k, delta, p_success, p_error)
print
print

print '# ZK''s equation'
# m is the total overhead symbols
for m in (0 .. 10):
    p_error = zk_p_err(k, m, q)
    p_success = 1 - p_error
    print 'In GF(%d), k=%d, delta=%d, p_success=%f, p_error=%f' % (q, k, m, p_success, p_error)



# for m in (0 .. 10):
#     print '# m = %d' % m
#     p_success = p(k=10, m=m, q=2)
#     p_fail = 1 - p_success
#     print '\tP_e = %f' % p_fail
#     print '\tP_s = %f' % p_success

# Lives equation
In GF(4), k=10, delta=0, p_success=0.688538, p_error=0.311462
In GF(4), k=10, delta=1, p_success=0.918050, p_error=0.081950
In GF(4), k=10, delta=2, p_success=0.979253, p_error=0.020747
In GF(4), k=10, delta=3, p_success=0.994797, p_error=0.005203
In GF(4), k=10, delta=4, p_success=0.998698, p_error=0.001302
In GF(4), k=10, delta=5, p_success=0.999675, p_error=0.000325
In GF(4), k=10, delta=6, p_success=0.999919, p_error=0.000081
In GF(4), k=10, delta=7, p_success=0.999980, p_error=0.000020
In GF(4), k=10, delta=8, p_success=0.999995, p_error=0.000005
In GF(4), k=10, delta=9, p_success=0.999999, p_error=0.000001
In GF(4), k=10, delta=10, p_success=1.000000, p_error=0.000000


# ZKs equation
In GF(4), k=10, delta=0, p_success=0.688538, p_error=0.311462
In GF(4), k=10, delta=1, p_success=0.918051, p_error=0.081949
In GF(4), k=10, delta=2, p_success=0.979255, p_error=0.020745
In GF(4), k=10, delta=3, p_success=0.994798, p_error=0.005202
In GF(4), k=10, delta=4, p_success=0.998700, p_error=0.001300
In GF(4), k=10, delta=5, p_success=0.999676, p_error=0.000324
In GF(4), k=10, delta=6, p_success=0.999920, p_error=0.000080
In GF(4), k=10, delta=7, p_success=0.999981, p_error=0.000019
In GF(4), k=10, delta=8, p_success=0.999996, p_error=0.000004
In GF(4), k=10, delta=9, p_success=1.000000, p_error=0.000000
In GF(4), k=10, delta=10, p_success=1.000000, p_error=0.000000