Abstract Algebra Theory and Applications - Computer Science ...

124 CHAPTER 7 ALGEBRAIC CODING THEORYWe leave the proof of this theorem as an exercise. In light of the theorem,the first n−m bits in x are called information bits and the last m bits arecalled check bits. In Example 12, the first three bits are the informationbits and the last three are the check bits.Theorem 7.8 Suppose that G is an n×k standard generator matrix. ThenC = {y : Gx = y for x ∈ Z k 2 } is an (n, k)-block code. More specifically, C isa group code.Proof. Let Gx 1 = y 1 and Gx 2 = y 2 be two codewords. Then y 1 + y 2 isin C sinceG(x 1 + x 2 ) = Gx 1 + Gx 2 = y 1 + y 2 .We must also show that two message blocks cannot be encoded into thesame codeword. That is, we must show that if Gx = Gy, then x = y.Suppose that Gx = Gy. ThenGx − Gy = G(x − y) = 0.However, the first k coordinates in G(x − y) are exactly x 1 − y 1 , . . . , x k − y k ,since they are determined by the identity matrix, I k , part of G. Hence,G(x − y) = 0 exactly when x = y.□Before we can prove the relationship between canonical parity-check matricesand standard generating matrices, we need to prove a lemma.Lemma ( 7.9 Let ) H = (A | I m ) be an m × n canonical parity-check matrixand G =In−mAbe the corresponding n×(n−m) standard generator matrix.Then HG = 0.Proof. Let C = HG. The ijth entry in C isc ij ===n∑h ik g kjk=1n−m∑h ik g kj +k=1n−m∑k=1= a ij + a ij= 0,a ik δ kj +n∑k=n−m+1n∑k=n−m+1h ik g kjδ i−(m−n),k a kj