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

\begin{document}

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

\begin{itemize}
    \item 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 $$C$$:
    $$C = \{x \in \mathbb{F}_q^n : Hx^T = 0\}.$$
    $$Hv^T$$ is called the syndrome of $$v$$ (with respect to $$H$$).

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

\paragraph{Definition 2.1.3 (Support of a word).} Let $$x = (x_1, \ldots, x_n) \in \mathbb{F}_q^n$$. 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, \ldots, x_n \rangle_{\mathbb{F}_q}.$$

\end{document}

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Here are some related questions:

1. What is a parity-check matrix, and why is it significant in coding theory?
2. What does it mean for a generator matrix $G$ to be in systematic form?
3. How is the syndrome of a vector used in error detection and correction?
4. What is the role of the support of a word in the context of vector spaces?
5. How do parity-check matrices and generator matrices interact in linear codes?

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