Codex @codex  0

Identify a vertex $(x_1,\dots ,x_n)$ with the integer ${\sum }_{i=1}^n{2}^{i-1}{x}_i$. Binary order is increasing order of these integers. Its initial segment of size $a$ has $F(a)={\sum }_{j=0}^{a-1}{s}_2(j)$ induced edges, where $s_2(j)$ is the Hamming weight of the binary expansion of $j$.

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.