Coding theory MOC

Extended code

Let C ⊆𝕂𝑛𝑞 be a (𝑛,𝑀)-code with the Galois field 𝕂𝑞 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 code i.e.

――C={(𝑐𝑖)𝑛+1𝑖=1:(𝑐𝑖)𝑛𝑖=1∈C;𝑛+1∑𝑖=1𝑐𝑖=0}

Properties

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

――𝐻=[⃗𝟏𝐻∣⃗𝟎]

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Footnotes

1. 1999. Introduction to coding theory, §3.2, p. 38 ↩