JavaReedSolomon | Backblaze Reed-Solomon Implementation in Java

I looked at the "simple RS" example "GF28" Java code. The decoder appears to handle erasures only (one of the inputs is an array of bad indices). It's using GF(256) based on hex 11B = x^8 + x^4 + x^3 + x + 1, the same as AES encryption. It is somewhat unusual choice since the lowest "primitive" is 3 (all numbers other than zero can be considered to be a power of 3), rather than the fields where the "primitive" is 2. The field polynomial is defined via PX, so it can be easily changed. I'm not sure why it generates tables dynamically instead of generating them during initialization, using a second set of true/false tables to indicate if specific table values have been generated.

I have a C RS demo program for 8 bit GF(256) fields. It's interactive, you select a field (there are 30 of them), whether to use a self reciprocal polynomial (it's usually not used), if the first consecutive root of the generator polynomial is 1 (if no is specified, then the first consecutive root is the "primitive"), number of parity bytes, and number of data bytes. It handles both erasures and errors, and since it's a demo program, it includes code for the 3 main types of decoder algorithms Peterson matrix algorithm, Sugiyama's adaptation of extended Euclid algorithm, and Berlekamp Massey algorithm, and also Forney algorithm to generate error values. The interactive part could be replaced with code that selects all these parameters. For your situation, change the define for MAXPAR from 20 to 80 (maximum number of parities). The user input is via stdin, so a text input file can be used to run the demo.

http://rcgldr.net/misc/eccdemo8.zip

In my demo, to generate a code word, the user enters values (user option "E" or "C"), then enters "N" to encode. To generate errors, the user enters values at specific locations. To generate erasures, the user uses the "P" option to enter erasure values at specific locations. "F" is used to fix (correct) the code word.

The wiki article includes the logic for the 3 main types of decoders. It also explains the Fourier transform, but the Fourier transform requires using one of the other decoders in order to generate the error polynomial, and isn't practical.

https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction#Error_correction_algorithms

Based on your comment I looked at the Zxing library. The method descriptions are a bit terse and use the wrong terminology. Here's a redo of the description: