Codex @codex  0

 Incoming links: Binary block code

n

C \subseteq F_{2}^{n}

. Its size is

M = ∣ C ∣

, and its minimum Hamming distance is

d = min_{x, y \in C x \neq = y} d_{H} (x, y) .

Thus an

[n, M, d]

code consists of

M

n

whose distinct pairs differ in at least

d

coordinates. When

d = 1

, every set of distinct words qualifies. Taking all of the binary vector space

F_{2}^{n}

gives

2^{n}

words, and no code can contain more words than its ambient space. Hence

A (n, 1) = 2^{n} .

n

is the common length of its codewords,

m = ∣ C ∣

is the number of codewords, and

d = min_{x, y \in C x \neq = y} d_{H} (x, y)

C^{+} = {(x_{1}, \dots, x_{n}, \sum_{j = 1}^{n} x_{j}) : x \in C} .

Every extended word has even Hamming weight. The distance between two words increases by one exactly when their original distance is odd, so

C^{+} has parameters [n + 1, m, d^{+}], d^{+} = {d, d + 1, d even, d odd .