Develop an ECC code for a 16bit data word Generate the code

Develop an ECC code for a 16-bit data word. Generate the code for the data word 0101 0000 0011 1001. Show that the code will correctly identify an error in data bit 5.

Solution

Parity positions are in 2ⁿ = 2⁰, 2¹, 2², 2³, 2⁴, 2⁵,……

                                            =1,2,4,8,16,32,………

16-bit data word is given as 0 1 0 1 0 0 0 0 0 0 1 1 1 0 0 1

Let D = D1, D2,D3,…. be data bits and P = P1, P2, P3,….be parity bits.

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P1

P2

D3

P4

D5

D6

D7

P8

D9

D10

D11

D12

D13

D14

D15

P16

D17

D18

D19

D20

D21

P1= write 1 number ,skip next number, write 1 number and so on….

   = bit positions (1,3,5,7,9,11,13,15,17,19,21)

    =P1,D3,D5,D7,D9,D11,D13,D15,D17,D19,D21

    =XOR (D3,D5,D7,D9,D11,D13,D15,D17,D19,D21)

    =XOR(0,1,1,0,0,0,1,1,0,1)

    = 1

P2= write 2 numbers in sequence from pos.2, skip two numbers in sequence, and so on….

     = bit positions (2,3,6,7,10,11,14,15,18,19)

    = P2,D3,D6,D7,D10,D11,D14,D15,D18,D19

     = XOR(D3,D6,D7,D10,D11,D14,D15,D18,D19)

    = XOR( 0,0,1,0,0,0,1,1,0)

= 1

P4= write 4 numbers in sequence from pos.4, skip next 4 numbers in sequence and so on…

   = bit positions (4,5,6,7,12,13,14,15,20,21)

=P4,D5,D6,D7,D12,D13,D14,D15,D20,D21

=XOR(D5,D6,D7,D12,D13,D14,D15,D20,D21)

= XOR (1,0,1,0,0,0,1,0,1)

= 0

P8= write 8 numbers in sequence from pos. 8 , skip next 8 numbers and so on…

    =bit positions(8,9,10,11,12,13,14,15)

   =   P8,D9,D10,D11,D12,D13,D14,D15

= XOR(D9,D10,D11,D12,D13,D14,D15)

=XOR(0,0,0,0,0,0,1)

=1

P16= write 16 numbers in sequence from pos. 16 , skip next 16 numbers and so on…

     = bit positions ( 16,17,18,19,20,21)

   =P16,D17,D18,D19,D20,D21

   =XOR(D17,D18,D19,D20,D21)

= XOR (1,1,0,0,1)

= 1

FINALLY è

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Check bits are evaluated asè

C1= XOR(P1,D3,D5,D7,D9,D11,D13,D15,D17,D19,D21)

    = XOR(1,0,1,1,0,0,0,1,1,0,1) = 0

C2 = = XOR(P2,D3,D6,D7,D10,D11,D14,D15,D18,D19)

    = XOR(1, 0,0,1,0,0,0,1,1,0) = 0

C4= =XOR(P4,D5,D6,D7,D12,D13,D14,D15,D20,D21)

= XOR (0,1,0,1,0,0,0,1,0,1) =0

C8 = = XOR(P8,D9,D10,D11,D12,D13,D14,D15)

=XOR(1,0,0,0,0,0,0,1) =0

C16 =XOR(P16,D17,D18,D19,D20,D21)

= XOR (1,1,1,0,0,1)=0

Since C = C16 C8 C4 C2 C1 = 00000 , THERE IS NO ERROR

Suppose there is error in bit 5

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Check bits are evaluated asè

C1= XOR(P1,D3,D5,D7,D9,D11,D13,D15,D17,D19,D21)

    = XOR(1,0,0,1,0,0,0,1,1,0,1) = 1

C2 = = XOR(P2,D3,D6,D7,D10,D11,D14,D15,D18,D19)

    = XOR(1, 0,0,1,0,0,0,1,1,0) = 0

C4= =XOR(P4,D5,D6,D7,D12,D13,D14,D15,D20,D21)

= XOR (0,0,0,1,0,0,0,1,0,1) =1

C8 = = XOR(P8,D9,D10,D11,D12,D13,D14,D15)

=XOR(1,0,0,0,0,0,0,1) =0

C16 =XOR(P16,D17,D18,D19,D20,D21)

= XOR (1,1,1,0,0,1)=0

Since C = C16 C8 C4 C2 C1 = 00101 , THERE IS ERROR in bit 5

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
P1	P2	D3	P4	D5	D6	D7	P8	D9	D10	D11	D12	D13	D14	D15	P16	D17	D18	D19	D20	D21