Abstract Algebra Theory and Applications - Computer Science ...

122 CHAPTER 7 ALGEBRAIC CODING THEORYand I m is the m × m identity matrix⎛1 0 · · · 00 1 · · · 0⎜⎝.. . .. .0 0 · · · 1⎞⎟⎠ .With each canonical parity-check matrix we can associate an n × (n − m)standard generator matrix( )In−mG = .AOur goal will be to show that Gx = y if and only if Hy = 0. Given amessage block x to be encoded, G will allow us to quickly encode it into alinear codeword y.Example 12. Suppose that we have the following eight words to be encoded:(000), (001), (010), . . . , (111).For⎛A = ⎝0 1 11 1 01 0 1the associated standard generator and canonical parity-check matrices areand⎛G =⎜⎝⎛H = ⎝1 0 00 1 00 0 10 1 11 1 01 0 1⎞⎠ ,⎞⎟⎠0 1 1 1 0 01 1 0 0 1 01 0 1 0 0 1respectively.Observe that the rows in H represent the parity checks on certain bitpositions in a 6-tuple. The 1’s in the identity matrix serve as parity checks⎞⎠ ,