Let $C\subset \mathbb{F}_3^3$ be a linear code with generator matrix $G=\begin{pmatrix} 1 & 2 & 0\\ 1 & 0 & 2 \end{pmatrix}$.
$(1)$ Find the standard form for the generator matrix and the partiy check matrix.
Answer: $G=[I_2|P]=\begin{pmatrix} 1 & 0 & 2\\ 0 & 1 & -1 \end{pmatrix}$ and thus $H=[-P^T|I_1]=\begin{pmatrix} -2 & 1 & 1\\ \end{pmatrix}$.
$(2)$ What is $d(C)$? [minimum distance of $C$]
Answer: I read online that it is the minimum number of linear dependent columns of the parity check matrix, that would be $d(C)=2$, right? Because of the second and third row being linearly dependent.
$(3)$ Show that $C$ is cyclic
Is there a way to show this using the matrices?
$(4)$ Find the generator and parity check polynomial of $C$.
$(5)$ Is $C$ a BCH-Code?
I have no attempts for $(4)$ and $(5)$ – any help would be nice!
Best Answer
(4) The generator polynomial of a linear code $C$ in $\mathbb F_q^n$ is a divisor of $x^n-1$ and has degree $n-k$, where $k$ is the dimension of $C$ as a vector space over $\mathbb F_q$. In your case, $q=n=3$ and $k=2$, so the generator polynomial $g(x)$ you're looking for divides $x^3-1=(x-1)^3$ and has degree $1$. The only option is $g(x)=x-1$. The parity check polynomial then is $h(x)=\frac{x^n-1}{g(x)}=(x-1)^2$.