Extended code

Let $C \subseteq 𝕂 𝑛 𝑞$ be a $(𝑛, 𝑀)$ -code with the Galois field $𝕂 𝑞$ as its alphabet. The corresponding extended code $―― C$ is a $(𝑛 + 1, 𝑀)$ -code defined by adding an additional parity check digit so that the sum of all digits is always zero,¹ #m/def/code i.e.

―― C = {(𝑐 𝑖) 𝑛 + 1 𝑖 = 1 : (𝑐 𝑖) 𝑛 𝑖 = 1 \in C; 𝑛 + 1 \sum 𝑖 = 1 𝑐 𝑖 = 0}

Properties

If $C$ is a linear code with parity check matrix $𝐻$ , then $―― C$ has parity check matrix

―― 𝐻 = [⃗ 𝟏 𝐻 ∣ ⃗ 𝟎]

#state/tidy | #lang/en | #SemBr

1999. Introduction to coding theory, §3.2, p. 38 ↩

Extended code

Properties

Footnotes