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1 Prove: Suppose C is a linear [n, k]-code having generator matrix G

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1 Problem sheet 6

2006 MT3610/5461 Error correcting codes

Ex. 1

Prove: Suppose C is a linear [n, k]-code having generator matrix G. Then a vector ~v ∈ V(n, q) belongs to C⊥ ⇔ ~vG> = 0, where G> denotes the transpose ofG.

Ex. 2

Determine the dual codes C1⊥ and C2⊥ for C1 = {000,111} and C2 = {00000,11111}.

Ex. 3

Prove that ifEn is the binary even weight code of length n, then En⊥ is the repetition code of lengtn. Give the generator matrices and the parity check matrices for the two codes.

Ex. 4

LetC be the ternary linear code with generator matrix

1 1 1 0

2 0 1 1

.

a) Find a generator matrixG forC in standard form b) Find a parity check matrix H forC in standard form.

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