Bch Code Coding Theory Sagemath

Leo Migdal

-Nov 17, 2025, 2:55 PM

Let $F = \GF{q}$ and $\Phi$ be the splitting field of $x^{n} - 1$ over $F$, with $n$ a positive integer. Let also $\alpha$ be an element of multiplicative order $n$ in $\Phi$. Finally, let $b, \delta, \ell$ be integers such that $0 \le b \le n$, $1 \le \delta \le n$ and $\alpha^\ell$ generates the multiplicative group $\Phi^{\times}$. A BCH code over $F$ with designed distance $\delta$ is a cyclic code whose codewords $c(x) \in F[x]$ satisfy $c(\alpha^{a}) = 0$, for all integers $a$ in the arithmetic sequence \(b, b + \ell,... Representation of a BCH code seen as a cyclic code. base_field – the base field for this code

designed_distance – the designed minimum distance of the code There was an error while loading. Please reload this page. There was an error while loading. Please reload this page. Author: David Joyner and Robert Miller (2008), edited by Ralf Stephan for the initial version.

David Lucas (2016) for this version. Marketa Slukova (2019) for the latest version. This tutorial, designed for beginners who want to discover how to use Sage for their work (research, experimentation, teaching) on coding theory, will present several key features of Sage’s coding theory library and explain... During this tutorial, we will cover the following parts: what can you do with generic linear codes and associated methods, what can you do with structured code families,

Coding theory is the mathematical theory for algebraic and combinatorial codes used for forward error correction in communications theory. Sage provides an extensive library of objects and algorithms in coding theory. Basic objects in coding theory are codes, channels, encoders, and decoders. The following modules provide the base classes defining them. Catalogs for available constructions of the basic objects and for bounds on the parameters of linear codes are provided. The following module is a base class for linear code objects regardless their metric.

There is a number of representatives of linear codes over a specific metric. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Ask questions, find answers and collaborate at work with Stack Overflow Internal. Ask questions, find answers and collaborate at work with Stack Overflow Internal. Explore Teams Connect and share knowledge within a single location that is structured and easy to search.

Let $\alpha $ be a root of $X^4 + X + 1$, and let C be the BCH code of length 15 with defining set the first four powers of $\alpha $. Determine the error position of the following received word: $1+X^6 + X^7 + X^8$. The named code families below are represented in Sage by their own classes, allowing specialised implementations of e.g. decoding or computation of properties: In contrast, for some code families Sage can only construct their generator matrix and has no other a priori knowledge on them: Sage supports the following derived code constructions.

If the constituent code is from a special code family, the derived codes inherit e.g. decoding or minimum distance capabilities: Other derived constructions that simply produce the modified generator matrix can be found among the methods of a constructed code. Enter search terms or a module, class or function name. Author: David Joyner and Robert Miller (2008), edited by Ralf Stephan for the initial version. David Lucas (2016) for this version.

Marketa Slukova (2019) for the latest version. This tutorial, designed for beginners who want to discover how to use Sage for their work (research, experimentation, teaching) on coding theory, will present several key features of Sage’s coding theory library and explain... During this tutorial, we will cover the following parts: what can you do with generic linear codes and associated methods, what can you do with structured code families, There was an error while loading.

Please reload this page. There was an error while loading. Please reload this page. One of the properties of BCH codes is that their dual is a cyclic code but sage doesn't return a cyclic code object on call of .dual_code(). Since BCH codes $\subseteq$ cyclic linear codes $\subseteq$ linear codes, I would expect to obtain a cyclic code on which I can call C.dual().generator_polynomial() for example. And since generator polynomials are not a thing for linear codes, the method is not exposed which makes sense.

Bch Code Coding Theory Sagemath

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