Y.S.Han Introduction to Binary Linear Block Codes 10 Example T100110 G= 010101 001011 be a generator matrix of C and 110100 H= 1 0 10 10 011001 be a parity-check matrix of C. c=uG School of Electrical Engineering Intelligentization,Dongguan University of Technology
Y. S. Han Introduction to Binary Linear Block Codes 10 Example G = 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1 be a generator matrix of C and H = 1 1 0 1 0 0 1 0 1 0 1 0 0 1 1 0 0 1 be a parity-check matrix of C. c = uG School of Electrical Engineering & Intelligentization, Dongguan University of Technology
Y.S.Han Introduction to Binary Linear Block Codes 11 u∈{000,100,010,001,110,101,011,111} C={000000,100110,010101,001011, 110011,101101,011110,111000X School of Electrical Engineering Intelligentization,Dongguan University of Technology
Y. S. Han Introduction to Binary Linear Block Codes 11 u ∈ {000, 100, 010, 001, 110, 101, 011, 111} C = {000000, 100110, 010101, 001011, 110011, 101101, 011110, 111000} School of Electrical Engineering & Intelligentization, Dongguan University of Technology
Y.S.Han Introduction to Binary Linear Block Codes 12 Parity-Check Matrix 1.A parity check for C is an equation of the form a0c0⊕a1C⊕..⊕an-1Cn-1=0, which is satisfied for any c=(co,c1,...,cn-1)EC. 2.The collection of all vectors a (ao,a1,...,an-1)forms a subspace of Vn.It is denoted by C+and is called the dual code of C. 3.The dimension of C+is n-k and C+is an (n,n-k)BLBC. Any generator matrix of C is a parity-check matrix for C and is denoted by HI. 4.CHT=01x(n-k)for any cEC. 5.Let G=IkA].Since cHT =uGHT=0,GHT must be 0.If School of Electrical Engineering Intelligentization,Dongguan University of Technology
Y. S. Han Introduction to Binary Linear Block Codes 12 Parity-Check Matrix 1. A parity check for C is an equation of the form a0 c0 ⊕ a1 c1 ⊕ . . . ⊕ an−1 cn−1 = 0, which is satisfied for any c = (c0, c1, . . . , cn−1) ∈ C. 2. The collection of all vectors a = (a0, a1, . . . , an−1) forms a subspace of V n. It is denoted by C⊥ and is called the dual code of C. 3. The dimension of C⊥ is n − k and C⊥ is an (n, n − k) BLBC. Any generator matrix of C⊥ is a parity-check matrix for C and is denoted by H. 4. cHT = 01×(n−k) for any c ∈ C. 5. Let G = [IkA]. Since cHT = uGHT = 0, GHT must be 0. If School of Electrical Engineering & Intelligentization, Dongguan University of Technology
Y.S.Han Introduction to Binary Linear Block Codes 13 H=[ArIn-]小，theu GHT-A]-ATL Thus,the above II is a parity-check matrix. 6.Let c be the transmitted codeword and y is the binary received vector after quantization.The vector e=cy is called an error pattern. 7.Lety=c⊕e. s=yHr=(c⊕e)HT=eHT which is called the syndrome of y. 8.Let S={ss=yHT for allyevn}be the set of all School of Electrical Engineering Intelligentization,Dongguan University of Technology
Y. S. Han Introduction to Binary Linear Block Codes 13 H = [ −A T In−k ] , then GHT = [IkA] [ −A T In−k ]T = [IkA] −A In−k = −A + A = 0k×(n−k) Thus, the above H is a parity-check matrix. 6. Let c be the transmitted codeword and y is the binary received vector after quantization. The vector e = c ⊕ y is called an error pattern. 7. Let y = c ⊕ e. s = yHT = (c ⊕ e)HT = eHT which is called the syndrome of y. 8. Let S = {s|s = yHT for all y ∈ V n } be the set of all School of Electrical Engineering & Intelligentization, Dongguan University of Technology
Y.S.Han Introduction to Binary Linear Block Codes 14 syndromes.Thus,S=2-k(This will be clear when we present standard array of a code later). School of Electrical Engineering Intelligentization,Dongguan University of Technology
Y. S. Han Introduction to Binary Linear Block Codes 14 syndromes. Thus, |S| = 2n−k (This will be clear when we present standard array of a code later) . School of Electrical Engineering & Intelligentization, Dongguan University of Technology