README for Reed-Solomon Decoder (Ling Li, Caltech, Feb. 5, 2001)
----------------------------------------------------------------------

EE/Ma 127b Project 1, a decoder for (31,15) R-S code over GF(32).
Implemented in Java.

Please see http://www.cs.caltech.edu/~ling/course/ee127b/ for more
about decoding R-S codes.

----------------------------------------------------------------------

Summary:
   Operations with numbers and polynomials over $GF(32)$ are needed.
   So I designed two classes. One (GF) deals with numbers in $GF(q)$
   and the other (GFPoly) with polynomials over $GF(q)$. GFPoly
   provides methods like euclid and fft.

   The class RS, by calling methods in GF and GFPoly, does the work of
   calculating the syndrome $S$, $\sigma$ and $\omega$, and decoding 
   in time domain and frequency domain. Please refer to RJM Chapter 9 
   for the details of the decoding.

   The input/output conversion is carried out in a Java applet. Either
   form, such as "A*B1*3" or "0,*,1,26,*,28" can be used for input.

Exceptions:
   1. Too many erasures. The number of erasures should not excel 16.
   2. Syndrome is 0. A valid codeword received.
   3. $\sigma_0$ is 0. $\sigma$ can't be `normalized'.
   4. Degree of $\sigma$ is less than or equal to that of $\omega$.
      Too many errors prevent us to get a correct $\sigma$ --- the
      degree of $\sigma$ should be larger than $(r-e_0)/2$.
   5. (Time domain) $\sigma$ has not enough roots. $\sigma$ should 
      has $d$ distinct roots if it is of degree $d$.
   6. (Frequency domain) $S$ verified error.

----------------------------------------------------------------------

Files:
   ecc/GF.java          +,-,*,/ with numbers in GF(q).
   ecc/GFPoly.java      +,-,*,/,euclid with polynomials over GF(q).
   ecc/RS.java          RS decoder.
   RSDecoder.java       The decoder applet.

----------------------------------------------------------------------

Problems:
   1. This is my first time to do project in Java. Ugly programming.
   2. I am not sure about RS when $n < p^m-1$. That is, for shortened 
      R-S codes, class RS may do wrong work.

----------------------------------------------------------------------

Note:
   Designed by JBuilder 3.5, JDK 1.2.2;
   In order to be compatible with browsers using JDK 1.0 (or 1.1?),
   the codes corresponding to GridBagLayout was manually revised.

----------------------------------------------------------------------
This file together with the applet and its programs can be found at
http://www.cs.caltech.edu/~ling/course/ee127b/proj1/