📘 coding theory

1. **State the problem:** Given a generator matrix $G$ and a message vector $\mathbf{m} = (m_3, m_2, m_1, m_0) = (1,0,1,0)$, find the code vector $\mathbf{c}$. 2. **Formula used:**

1. **Problem Statement:** Find the code words generated by the parity check matrix $$H=\begin{pmatrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

1. **Problem Statement:** Find the code words generated by the parity check matrix $$H=\begin{pmatrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}$$ w

1. Let's start by defining the minimum distance $d_{min}$ of a code. It is the smallest Hamming distance between any two distinct codewords in the code.\n\n2. The minimum distance

1. **Problem statement:** Find all possible binary cyclic codes of length 7, determine their minimum distance, and check if they are perfect. 2. **Background:** A binary cyclic cod