Contents

• Contents
• Hamming codes
• Reed-Muller codes
• Cyclic codes
• Bibliography
• About this document ...

Let $\mathbb{F}$ be a finite field. Consider a short exact sequence of vector spaces

$$0 \rightarrow \mathbb{F}^k \stackrel{\gamma}{\rightarrow} \... ...\stackrel{\theta}{\rightarrow} \mathbb{F}^{n-k} \rightarrow 0.$$

A linear code is the image of $\gamma$. Since the sequence is exact, a vector $v\in \mathbb{F}^n$ is a codeword if and only if $\theta(v)=0$. If $\mathbb{F}^i$ is given the usual standard vector space basis then the matrix of $\gamma$ is the generating matrix and the matrix of $\theta$ is the parity check matrix.