https://github.com/wchresta/linear-code

Haskell library for linear codes from coding theory
https://github.com/wchresta/linear-code

algebra coding-theory linear-codes math mathematics

Last synced: 2 months ago
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Haskell library for linear codes from coding theory

• Host: GitHub
• URL: https://github.com/wchresta/linear-code
• Owner: wchresta
• License: gpl-3.0
• Created: 2018-06-22T11:53:43.000Z (over 6 years ago)
• Default Branch: master
• Last Pushed: 2018-08-30T06:26:33.000Z (over 6 years ago)
• Last Synced: 2024-09-30T07:09:55.613Z (3 months ago)
• Topics: algebra, coding-theory, linear-codes, math, mathematics
• Language: Haskell
• Size: 73.2 KB
• Stars: 4
• Watchers: 3
• Forks: 0
• Open Issues: 1
• Metadata Files:
  • Readme: README.md
  • Changelog: ChangeLog.md
  • License: LICENSE

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# linear-code

Library to handle linear codes from coding theory.

The library is designed to carry the most important bits of information in the

type system while still keeping the types sane.

This library is based roughly on [_Introduction to Coding Theory_ by _Yehuda Lindell_](http://u.cs.biu.ac.il/~lindell/89-662/coding_theory-lecture-notes.pdf)

# Usage example

## Working with random codes

```Haskell

> :m + Math.Code.Linear System.Random

> :set -XDataKinds

> c <- randomIO :: IO (LinearCode 7 4 F5)

> c

[7,4]_5-Code

> generatorMatrix c

( 1 0 1 0 0 2 0 )

( 0 2 0 0 1 2 0 )

( 0 1 0 1 0 1 0 )

( 1 0 0 0 0 1 1 )

> e1 :: Vector 4 F5

( 1 0 0 0 )

> v = encode c e1

> v

( 1 0 1 0 0 2 0 )

> 2 ^* e4 :: Vector 7 F3

( 0 0 0 2 0 0 0 )

> vWithError = v + 2 ^* e4

> vWithError

( 1 0 1 2 0 2 0 )

> isCodeword c v

True

> isCodeword c vWithError

False

> decode c vWithError

Just ( 1 0 2 2 2 2 0 )

```

Notice, the returned vector is NOT the one without error. The reason for this

is that a random code most likely does not have a distance >2 which would be

needed to correct one error. Let's try with a hamming code

## Correcting errors with hamming codes

```Haskell

> c = hamming :: BinaryCode 7 4

> generatorMatrix c

( 1 1 0 1 0 0 0 )

( 1 0 1 0 1 0 0 )

( 0 1 1 0 0 1 0 )

( 1 1 1 0 0 0 1 )

> v = encode c e2

> vWithError = v + e3

> Just v' = decode c vWithError

> v' == v

True

```