https://github.com/jakewilliami/codingtheory.jl

Pure Julia implementation of tools used in Coding Theory
https://github.com/jakewilliami/codingtheory.jl

algorithms codewords coding coding-theory julia julia-language julia-package julialang math mathematics maths

Last synced: 13 days ago
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Pure Julia implementation of tools used in Coding Theory

• Host: GitHub
• URL: https://github.com/jakewilliami/codingtheory.jl
• Owner: jakewilliami
• License: mit
• Created: 2020-10-05T21:21:03.000Z (about 4 years ago)
• Default Branch: master
• Last Pushed: 2024-10-06T04:38:54.000Z (about 1 month ago)
• Last Synced: 2024-10-14T16:43:23.996Z (22 days ago)
• Topics: algorithms, codewords, coding, coding-theory, julia, julia-language, julia-package, julialang, math, mathematics, maths
• Language: Julia
• Homepage:
• Size: 374 KB
• Stars: 5
• Watchers: 2
• Forks: 1
• Open Issues: 2
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

        


    CodingTheory.jl



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This is a minimal package for a pure Julia implementation of tools used in [Coding Theory](https://en.wikipedia.org/wiki/Coding_theory).  This is the science of accurately transmitting information through a noisy channel.

## Background

We assume that Alice and Bob communicate by sending sequences of symbols from a finite set *Σ*, which we call the **alphabet**.  We always use *q* to stand for the size of the set of symbols, |*Σ*|.  A **word** is a sequence of symbols from the alphabet *Σ*.  If *w1w2...wn* is such a word, then *n* is its **length**.  We use *Σn* to denote the set of words with length *n* using symbols in *Σ*.  In general, the number of words in *Σn* is



    |Σn| = qn.



**Block codes** are codes in which Alice transmits words of a preditermined and fixed length.  A **code** is a subset *C ⊆ Σn*.  The words in *C* are called **code words**.  We say that *n* is the **block length**.  We use *M* to stand for *|C|*, the number of code words.  Alice has a set *ℳ*, some of which she wants to send to Bob, so she has the bijective encoding function



    E : ℳ ⟶ C.



Similarly, Bob has a decoding function



    D : Σn ⟶ C ∪ {?},



Where Bob uses the *?* symbol when he cannot confidently decode.  So if Alice wishes to communicate a message, she transmits a code word *w = E(M)*.  *w* may be corrupted to *w' ≠ w*.  Then Bob can decode *w'* as *E-1(D(w))*.  If Bob is not certain how to decode, then *D(w')* may be '*?*', which means that Bob can tell an error has occurred but is not certain what that error is.

If *ℳ ⊆ Σk* is the set of messages, then *k* is the **message length**.

## A note on the number of codewords in a code

We have some algorithms brute-force searching for the codewords in a [q, n, d]-code.  These algorithms are brute-force as they do not assume that q is a prime power.  Therefore, they go through all possible codewords of a [q, n]-code, and narrow down the code based on d.  There algorithms are namely `get_codewords_greedy` and `get_codewords_random`, both of which using `get_all_codewords`.  The `get_codewords` function iterates through possibilities of `get_codeword_random` and chooses the maximum of those iterations or the `get_codeword_greedy` length.  Despite the name, `get_codewords` is only a **probably** candidate.  Increate the keyword argument `m` to decrease the likelihood that there is a code with more codewords while maintaining the bound of the distance.  Furthermore, there is a `get_codewords` method that lists all linear combinations of rows of a generator matrix.

## Similar Projects

  - [Eric Sabo's CodingTheory](https://github.com/esabo/CodingTheory) may offer more comprehensive support for coding theory computation

  - [Claus Fieker, William Hart, Tommy Hofmann, and Fredrik Johansson's Nemo/Hecke](https://github.com/thofma/Hecke.jl) provides useful computer algebra and number theory implementations within Julia that are used throughout within this package.

## Citation

If your research depends on CodingTheory.jl, please consider giving us a formal citation: [`citation.bib`](./citation.bib).