CSC 310, ASSIGNMENT #4 - RESULTS AND DISCUSSION


Here is the output of the program:


                 Results:  all-ok-count/fraction  message-ok-count/fraction

errs  tries |       Method 1      |       Method 2      |       Method 3      |
   0    153 | 153/1.000  153/1.000| 153/1.000  153/1.000| 153/1.000  153/1.000|
   1    696 | 696/1.000  696/1.000| 696/1.000  696/1.000| 696/1.000  696/1.000|
   2   1514 |1514/1.000 1514/1.000|1514/1.000 1514/1.000|1514/1.000 1514/1.000|
   3   2012 |2012/1.000 2012/1.000|2012/1.000 2012/1.000|2012/1.000 2012/1.000|
   4   2073 |1965/0.948 2003/0.966|2058/0.993 2058/0.993|2073/1.000 2073/1.000|
   5   1624 |1337/0.823 1447/0.891|1565/0.964 1565/0.964|1620/0.998 1620/0.998|
   6    989 | 569/0.575  705/0.713| 848/0.857  850/0.859| 963/0.974  963/0.974|
   7    564 | 190/0.337  293/0.520| 391/0.693  394/0.699| 519/0.920  519/0.920|
   8    238 |  31/0.130   70/0.294| 110/0.462  110/0.462| 199/0.836  199/0.836|
   9     97 |   2/0.021   18/0.186|  19/0.196   20/0.206|  52/0.536   52/0.536|
>=10     40 |   0/0.000    2/0.050|   1/0.025    1/0.025|  10/0.250   10/0.250|
total 10000 |8469/0.847 8913/0.891|9367/0.937 9373/0.937|9811/0.981 9811/0.981|

Method 1 is row/col decoding, Method 2 is row/col/row/col, Method 3 is optimal


Your results will not necessarily be exactly the same, if the pseudo-random 
numbers you used were different.  The accuracy of the proportions of correct
decoding shown will vary with the number of messages that they are based on,
and on the proportion.  For the line with 8 errors, based on 238 messages, the
accuracy of the figure of 0.462 for the fraction of correct decoding using
Method 2 is accurate to around plus-or-minus 2*sqrt(0.462*(1-0.462)/238)=0.06
(with 95% confidence).  Most of the other proportions will be more accurate.
Since the methods were all tested on the same messages, the differences seen
between the methods may be more accurate than the individual figures.

We can see that all the methods decode correctly when there are no more than
three transmission errors.  The optimal method also always decodes correctly 
when there are four transmission errors, which is expected, since we know that
the minimum distance for this code is d=9, and hence it should be guaranteed
to correct all patterns of (d-1)/2 = 4 or fewer errors if decoded optimally.

For larger numbers of errors, all the methods still often decode correctly.  
Method 3 is best, as expected, and Method 2 is better than Method 1 (the 
apparent slight exception for >=10 errors is almost certainly a chance result 
in which Method 1 got lucky for the extra transmission in which it decoded 
the message bits correctly).

Method 1 and Method 2 sometimes decode the message bits correctly but don't 
get all the check bits correct.  We can conclude from this that these decoding 
methods don't always produce codewords as their decoding, since a codeword 
(by definition) has the right check bits to go with its message bits.  Method 3
always produces a codeword, since it works by finding the codeword closest 
to the received data.  Hence Method 3 can't possible get the message bits 
right without getting the check bits right too.

Method 2 seems like a good practical solution, since it improves on Method 1
by a non-neglibible amount, while being much faster than Method 3.