Notice
1.4. Parity Checking
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Descriptif
There are two standardways to describe a subspace, explicitly by giving abasis, or implicitly, by the solution space of the set ofhomogeneous linear equations. Therefore, there are twoways of describing a linear code, explicitly, as wehave seen in the previous sequence, by a generatormatrix, or implicitly, by the null space of a matrix. This iswhat we will see in this sequence. This leads to thefollowing definition: H is a parity check matrix of a linear code, ifthe code is the null space of H. In this way, any linear codeis completely specified by a
parity check matrix. Suppose that we have amessage of 4 bits, then we put them in the middle of theVenn Diagram, and we complete the empty three areasaccording to the following rules. The number of ones inevery circle is even. This gives us three redundantbits that we add to the message. We say that our circle isincorrect, if its parity is odd. So, if you modify anysymbol, then the error is propagated to the circlewhere this symbol is involved. Thus, if the circle one andthe second circle are wrong, then we deduce that an error hasoccurred in the first position. If the three circlesare wrong, then we deduce that there has been anerror in the fourth position.
Thus, the redundantinformation can be obtained from the message by means of threelinear equations or parity check.
Intervention
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