To convert the text in the image into code, I'll transcribe the mathematical notations in LaTeX, which is the standard format for representing mathematical documents. Here's the LaTeX code for the content in your image:

latexCopy code
\documentclass{article}
\usepackage{amsmath, amssymb}

\begin{document}

\section*{Code Definitions and Properties}

$$C = \{ xG, x \in \mathbb{F}_q^k \}$$

- by a parity-check matrix $$H \in \mathbb{F}_q^{(n-k) \times n}$$. Each row of $$H$$ determines a parity-check equation verified by the elements of $$C$$:
$$C = \{ x \in \mathbb{F}_q^n : Hx^T = 0 \}.$$

- $$Hx^T$$ is called the \textit{syndrome} of $$x$$ (with respect to $$H$$).

We say that $$G$$ (respectively $$H$$) is under systematic form if and only if it is of the form 
$$\begin{bmatrix}
I_k | A
\end{bmatrix}$$
(respectively 
$$\begin{bmatrix}
I_{n-k} | B
\end{bmatrix}
\)).

\subsection*{Definition 2.1.3 (Support of a word)}
Let $$x = (x_1, \dots, x_n) \in \mathbb{F}_q^n$$. The support $$E$$ of $$x$$, denoted $$\text{Supp}(x)$$, is the $$\mathbb{F}_q$$-subspace of $$\mathbb{F}_q^m$$ generated by the coordinates of $$x$$:

\[
E = \langle x_1, \dots, x_n \rangle_{\mathbb{F}_q},$$

and we have 
$$\dim E = \|x\|.$$

The number of supports of dimension $$w$$ of $$\mathbb{F}_q^m$$ is denoted by the Gaussian coefficient:

$$\binom{m}{w}_q.$$

\end{document}

Notes:

1. \mathbb{F}_q represents the finite field $\mathbb{F}_q$.
2. \langle x_1, \dots, x_n \rangle_{\mathbb{F}_q} denotes the subspace generated by the coordinates $x_1, \dots, x_n$.
3. The Gaussian coefficient $\binom{m}{w}_q$ is a standard notation in coding theory.

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Related Questions:

1. What are parity-check matrices, and how are they used in error detection?
2. How is the concept of a "syndrome" applied in coding theory?
3. What does "systematic form" mean in the context of matrices $G$ and $H$?
4. Can you explain the Gaussian coefficient $\binom{m}{w}_q$ and its significance?
5. How is the support of a word $x$ connected to its weight $\|x\|$?

Tip:

When working with LaTeX for mathematical content, always use the amsmath package to access advanced equation formatting tools.