Code Word Polynomial

A #Cyclic Code can be transformed into a Code Word Polynomial with the following equation:

$$ \begin{align} x &= (x_0, x_1, x_2, \ldots, x_{n-1})\\ x(D) &= x_0 + x_1 D + x_2 D^2 + \dots + x_{n-1} D^{n-1} \end{align} $$

Where:

$x$ is the code word
$x_n$ is $n$th position bit which could be 0 or 1
$D$ is an arbitrary real variable

For example, if $x = 10110$, we can transform it into polynomial $x(D) = 1 + D^2 + D^3$. As we can see, if $D^n$ exists, this means that there is a bit on the $n$th position.

Note: Addition in binary code is done with XOR and multiplication is done with AND.

Links to this page

TIT3131 Chapter 4: Cyclic Code

Code Word Polynomial
Syndrome Decoding

$R(D)$ is the received message polynomial#
Generator Polynomial

$D$ is an arbitrary real variable (see #Code Word Polynomial)

Generator Polynomial $g(D)$ is equivalent to the Generator Matrix in Linear Block Coding sense. Its degree is equal to the number of parity bits in the code which is $n-k$. It is unique as it is the only code word polynomial of minimum degree $n-k$, and any multiplication done on it is a #Code Word Polynomial.
Cyclic Code

We can describe a Cyclic Code of length $n$ with a polynomial of degree $n-1$, known as code word polynomial#. A Systematic# Cyclic Code should be in form of:

#math