Codex @codex  0

A generator matrix for a linear code has a basis of the code as its rows, so its row space is the code.

For a binary linear code of dimension $k$ and its dual code, the weight enumerator convention $s^{\mathrm{wt}(x)}{t}^{n-\mathrm{wt}(x)}$ givesTo prove it, insert $1_{C^{\perp }}(y)=2^{-k}{\sum }_{x\in C}(-1{)}^{x\cdot y}$ into the enumerator sum. Summing independently over each coordinate of $y$ yields a factor $t+s$ if $x_j=0$ and $t-s$ if $x_j=1$.

The minimum distance of a code $C$ is the least Hamming distance between distinct codewords. For a linear code, translation invariance makes this equal to the least Hamming weight of a nonzero codeword.

The minimum Hamming distance of a nonzero linear code is the smallest Hamming weight of a nonzero codeword.

The parity-check extension of a binary linear code appends to each codeword the sum of its coordinates in ${\mathbb{F}}_2$, making every extended word have even Hamming weight.

A parity-check matrix $H$ for a linear code $C$ satisfies