Parity Check Matrix

Let be a Linear Code A parity check matrix is a Generator Matrix for It is a matrix Note that

Lemma

An Linear Code with Generator Matrix has parity check matrix

Proof

Since So rows of generate a subcode of But Thus the rows of are a basis for as required.

Lemma

Let be a Linear Code with parity check matrix . Then if and only if:

1. any columns of are linearly independent
2. a set of columns of are linearly dependent.

Proof

Suppose has length Then If has columns Then if and only if