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Proof of concept on how to implement the Reed Solomon class of error correcting codes in Python

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# Reed Solomon Encoder and Decoder written in pure Python

Written from scratch by Andrew Brown <brownan@gmail.com> <brownan@cs.duke.edu> (c) 2010

I wrote this code as an exercise in implementing the Reed-Solomon error correction algorithm. This code is published in the hopes that it will be useful for others in learning how the algorithm works. (Nothing helps me learn something better than a good example!)

My goal was to implement a working Reed-Solomon encoder and decoder in pure python using no non-standard libraries. I also aimed to keep the code fairly well commented and organized.

However, a lot of the math involved is non-trivial and I can't explain it all in my comments. To learn more about the algorithm, see these resources:

The last two resources are course notes from Bruce Maggs' class, which I took this past semester. Those notes were immensely useful and should be read by anyone wanting to learn the algorithm.

Last two at Dr. Maggs' old site:

Also, here's a copy of the presentation I gave to the class Spring 2010 on my experience implementing this. The LaTeX source is in the presentation directory.

http://www.cs.duke.edu/courses/spring10/cps296.3/decoding_rs.pdf

## Files

rs.py
Holds the Reed-Solomon Encoder/Decoder object
polynomial.py
Contains the Polynomial object
ff.py
Contains the GF256int object representing an element of the GF(2^8) field

## Documentation

rs.RSCoder(n, k)

Creates a new Reed-Solomon Encoder/Decoder object configured with the given n and k values. n is the length of a codeword, must be less than 256 k is the length of the message, must be less than n

The code will have error correcting power s where 2s = n - k

The typical RSCoder is RSCoder(255, 223)

RSCoder Objects

RSCoder.encode(message, poly=False)

Encode a given string with reed-solomon encoding. Returns a byte string with the k message bytes and n-k parity bytes at the end.

If a message is < k bytes long, it is assumed to be padded at the front with null bytes.

The sequence returned is always n bytes long.

If poly is not False, returns the encoded Polynomial object instead of the polynomial translated back to a string (useful for debugging)

RSCoder.decode(r, nostrip=False)

Given a received string or byte array r, attempts to decode it. If it's a valid codeword, or if there are no more than (n-k)/2 errors, the message is returned.

A message always has k bytes, if a message contained less it is left padded with null bytes. When decoded, these leading null bytes are stripped, but that can cause problems if decoding binary data. When nostrip is True, messages returned are always k bytes long. This is useful to make sure no data is lost when decoding binary data.

RSCoder.verify(code)
Verifies the code is valid by testing that the code as a polynomial code divides g returns True/False

Besides the main RSCoder object, two other objects are used in this implementation. Their use is not specifically tied to the coder.

polynomial.Polynomial(coefficients=(), **sparse)

There are three ways to initialize a Polynomial object. 1) With a list, tuple, or other iterable, creates a polynomial using the items as coefficients in order of decreasing power

2) With keyword arguments such as for example x3=5, sets the coefficient of x^3 to be 5

3) With no arguments, creates an empty polynomial, equivalent to Polynomial((0,))

```>>> print Polynomial((5, 0, 0, 0, 0, 0))
5x^5```
```>>> print Polynomial(x32=5, x64=8)
8x^64 + 5x^32```
```>>> print Polynomial(x5=5, x9=4, x0=2)
4x^9 + 5x^5 + 2```

Polynomial objects export the following standard functions that perform the expected operations using polynomial arithmetic. Arithmetic of the coefficients is determined by the type passed in, so integers or GF256int objects could be used, the Polynomial class is agnostic to the type of the coefficients.

```__add__
__divmod__
__eq__
__floordiv__
__hash__
__len__
__mod__
__mul__
__ne__
__neg__
__sub__
evaluate(x)
degree()
Returns the degree of the polynomial
get_coefficient(degree)
Returns the coefficient of the specified term
```
ff.GF256int(value)
Instances of this object are elements of the field GF(2^8) Instances are integers in the range 0 to 255 This field is defined using the irreducable polynomial x^8 + x^4 + x^3 + x + 1 and using 3 as the generator for the exponent table and log table.

The GF256int class inherits from int and supports all the usual integer operations. The following methods are overridden for arithmetic in the finite field GF(2^8)

```__add__
__div__
__mul__
__neg__
__pow__
__rdiv__
__rmul__
__rsub__
__sub__
inverse()
Multiplicative inverse in GF(2^8)
```

## Examples

```>>> import rs
>>> coder = rs.RSCoder(20,13)
>>> c = coder.encode("Hello, world!")
>>> print repr(c)
'Hello, world!\x8d\x13\xf4\xf9C\x10\xe5'
>>>
>>> r = "\0"*3 + c[3:]
>>> print repr(r)
'\x00\x00\x00lo, world!\x8d\x13\xf4\xf9C\x10\xe5'
>>>
>>> coder.decode(r)
'Hello, world!'```

### Image Encoder

imageencode.py is an example script that encodes codewords as rows in an image. It requires PIL to run.

Usage: python imageencode.py [-d] <image file>

Without the -d flag, imageencode.py will encode text from standard in and output it to the image file. With -d, imageencode.py will read in the data from the image and output to standard out the decoded text.

An example is included: `exampleimage.png`. Try decoding it as-is, then open it up in an image editor and paint some vertical stripes on it. As long as no more than 16 pixels per row are disturbed, the text will be decoded correctly. Then draw more stripes such that more than 16 pixels per row are disturbed and verify that the message is decoded improperly.

Notice how the parity data looks different--the last 32 pixels of each row are colored differently. That's because this particular image contains encoded ASCII text, which generally only has bytes from a small range (the alphabet and printable punctuation). The parity data, however, is binary and contains bytes from the full range 0-255. Also note that either the data area or the parity area (or both!) can be disturbed as long as no more than 16 bytes per row are disturbed.

Proof of concept on how to implement the Reed Solomon class of error correcting codes in Python

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