Crandi

Polynomial Codes Bradencarlson Github Io

Leo Migdal
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polynomial codes bradencarlson github io

We can easily identify messages from $\mathbb{Z}_2^{k}$ with a polynomial in $\mathbb{Z}_2[x]$ with the following map \[ (a_0,a_1,\dots,a_{k-1})\rightarrow a_0+a_1x+\cdots +a_{k-1}x^{k-1} \] If we then fix some nonconstant polynomial $g(x)\in\mathbb{Z}_2[x]$ of degree $n-k$, we can... We will make use of Polynomial Rings in SageMath to implement this here. Consider the following example, taken from Judson's Abstract Algebra: Theory and Applications. Here is another example, complete with creating a message and encodeding it. Here we will expound on the polynomial codes that we explored in the previous document. BCH codes are very similar, although we are a little more careful with the choice of the generating polynomial.

Let $d=2r+1$. Then if \[ g(x)=\text{lcm}\{m_1(x),m_2(x),\dots,m_{2r}(x)\} \] where $m_i(x)$ is the minimal polynomial of $\omega^i$ where $\omega $ is the primitive $n^{th}$ root of unity over $\mathbb{Z}_2$, then with this choice of $g$, we have that... If we recall the previous results from the Linear codes that were discussed earlier, we have that this code can detect $2r$ errors, and correct $r$ errors. Here is an example of defining a BCH code, and encoding a message using the corresponding matrix. Similarly to what we did with Linear Codes, we define a BurstChannel that we can transmit our messages through, which will randomly flip bits in the message. This class will also have a higher probability of burst errors, that is, if the previous bit has been flipped, then the probability that the next bit will be flipped is higher.

In this cell we also define encode and decode functions, so that we may use lists for our messages, rather than matrices. Run the next cell to define this class, then proceed to the examples. There was an error while loading. Please reload this page. This project was developed for Math 4220 and Math 4230 at Southern Utah University, to provide students studying Abstract Algebra an approachable interface to SageMath, to help them understand introductory topics in the course. The goal of this project is to help students everywhere - although it is primarily aimed at students of Abstract Algebra - to learn the syntax of SageMath, so as to provide them with...

Please note that the layout of all of the files and sites closely follow the layout of the Abstract Algebra course videos given by Dr. Andrew Misseldine which can be found here. We assume that the student has no prior knowledge of the Python programming language, and has not used SageMath before. Therefore, the introduction documents may not be suitable for all users. If a student has prior experience with the Python programming language, they should jump right into the project. At any time during the use of this project, if you encounter an error in the preloaded code, or there are other issues that you find, or even if you have an idea on...

Here we will talk about linear codes, and how to analize them using SageMath. As discussed in Dr. Andrew Misseldine's lecture, we are interested in block codes, which will allow us to encode messages, as well as decode them and detect and potentially correct any errors (up to a certain number). The code to create the necesary matrices and message spaces is shown below. Now we create a Bernoulli class, which is a implementation of a Bernoulli distribution, which will then be used in conjunction with the SymmetricChannel class we define later, to introduce random errors into our... We chose the Bernoulli Distribution in this case because of our assumption that errors are uniformly distributed throughout the message, which in practice, is not accurate.

For a discussion of how to deal with errors that tend to occur in bursts, rather than uniformly, we will need the assistance of Polynomial Codes, to be discussed later. Here we go through an example of using the above classes to simulate transmitting a message through a Binary Symmectric Channel, which will introduce random errors into our message, then using our parity check... Run the cell above several times to see errors randomly appear in the recieved message, since our encoding matrix $G$ contains the identity, the first three bits of the recieved message are the original... We will now explore Rings in SageMath. There are a few rings that we have encountered in this project already, namely: The results here are different since SageMath knows to treat these are elements of different rings.

Here is another example. We can also form quotient rings within SageMath, see the following example, where we form the quotient ring \[ \frac{\mathbb{Z}[x]}{x^5-3x+1} \] Note that the indeterminant that we adjoint here is important, as it is how SageMath recognizes that a polynomial is an element of the Polynomial Ring. One last thing that we will do with rings in SageMath, we will do some calculations within a group ring. Recall that if $R$ is a ring and $G$ is a group, then the group ring $R(G)$ is defined to be \[ R(G)=\left\{\sum_{g\in G}\alpha_gg:\alpha_g\in R\right\} \] That is, $R(G)$ is the set of all... We demonstrate SageMath's ability to handle these in the following cell.

There was an error while loading. Please reload this page. There was an error while loading. Please reload this page. Here we go over a brief inroduction to SageMath, and its syntax, to provide us with the tools that we need to study some Abstract Algebra. Let's take a look at the following cell.

We can easily define variables, perform various operations on them, and have them printed out to the screen. Also note the optional use of f-strings. This is a feture inherited from Python, when a string is preceeded by an f, then any variable that appears in backets will have it's value printed, rather than the variable name. We can also combine variables to form other variables in SageMath. This will be very useful as we explore more complicated examples throughout this series. We will also need to know a little about types of data.

There are many different types of data that we can define, and we have already seen a few! consider the following cell. We see that not only do x and y have different values, but they are of different types. One is an integer, the other a string. Sage will tell us if something is wrong with our code, let's see an example.

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We Can Easily Identify Messages From $\mathbb{Z}_2^{k}$ With A Polynomial

We can easily identify messages from $\mathbb{Z}_2^{k}$ with a polynomial in $\mathbb{Z}_2[x]$ with the following map \[ (a_0,a_1,\dots,a_{k-1})\rightarrow a_0+a_1x+\cdots +a_{k-1}x^{k-1} \] If we then fix some nonconstant polynomial $g(x)\in\mathbb{Z}_2[x]$ of degree $n-k$, we can... We will make use of Polynomial Rings in SageMath to implement this here. Consider the following example, taken fro...

Let $d=2r+1$. Then If \[ G(x)=\text{lcm}\{m_1(x),m_2(x),\dots,m_{2r}(x)\} \] Where $m_i(x)$ Is

Let $d=2r+1$. Then if \[ g(x)=\text{lcm}\{m_1(x),m_2(x),\dots,m_{2r}(x)\} \] where $m_i(x)$ is the minimal polynomial of $\omega^i$ where $\omega $ is the primitive $n^{th}$ root of unity over $\mathbb{Z}_2$, then with this choice of $g$, we have that... If we recall the previous results from the Linear codes that were discussed earlier, we have that this code can detect $2r$ errors, and correct $...

In This Cell We Also Define Encode And Decode Functions,

In this cell we also define encode and decode functions, so that we may use lists for our messages, rather than matrices. Run the next cell to define this class, then proceed to the examples. There was an error while loading. Please reload this page. This project was developed for Math 4220 and Math 4230 at Southern Utah University, to provide students studying Abstract Algebra an approachable int...

Please Note That The Layout Of All Of The Files

Please note that the layout of all of the files and sites closely follow the layout of the Abstract Algebra course videos given by Dr. Andrew Misseldine which can be found here. We assume that the student has no prior knowledge of the Python programming language, and has not used SageMath before. Therefore, the introduction documents may not be suitable for all users. If a student has prior experi...

Here We Will Talk About Linear Codes, And How To

Here we will talk about linear codes, and how to analize them using SageMath. As discussed in Dr. Andrew Misseldine's lecture, we are interested in block codes, which will allow us to encode messages, as well as decode them and detect and potentially correct any errors (up to a certain number). The code to create the necesary matrices and message spaces is shown below. Now we create a Bernoulli cl...