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????  Digital Systems 
Department of Computer Science and Information
Engineering, Chaoyang University of
Technology ????????? Speaker Fuw-Yi Yang
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Text Book Digital Design 4th Ed.
Chapter 7 Hamming Code

One of the most common error-correcting codes
used in RAMs was devised by R. W. Hamming. In the
Hamming code, k parity bits are added to an n-bit
data word, forming a new word of n k
bits. Richard Wesley Hamming (Chicago, February
11, 1915 Monterey,
California, January 7, 1998)
was an American mathematician whose work had
many implications for computer science and
telecommunications. His contributions include the
Hamming code (which makes use of a Hamming
matrix), the Hamming window (described in Section
5.8 of his book Digital Filters), Hamming
numbers, Sphere-packing (or hamming bound) and
the Hamming distance.
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Text Book Digital Design 4th Ed.
Chapter 7 Hamming Code
Bit Position 1 2 3 4 5 6 7 8 9 10 11 12 Notation P1 P2 B3 P4 B5 B6 B7 P8 B9 B10 B11 B12 Data 1 1 0 0 0 1 0 0 Transmitted data P1 Xor of bits (3, 5, 7, 9, 11) P2 Xor of bits (3, 6, 7, 10, 11) P4 Xor of bits (5, 6, 7, 12) P8 Xor of bits (9, 10, 11, 12)

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Text Book Digital Design 4th Ed.
Chapter 7 Hamming Code
Bit Position 1 2 3 4 5 6 7 8 9 10 11 12 Notation P1 P2 B3 P4 B5 B6 B7 P8 B9 B10 B11 B12 Data 1 1 0 0 0 1 0 0 Received data C1 Xor of bits (1, 3, 5, 7, 9, 11) C2 Xor of bits (2, 3, 6, 7, 10, 11) C4 Xor of bits (4, 5, 6, 7, 12) C8 Xor of bits (8, 9, 10, 11, 12)

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Text Book Digital Design 4th Ed.
Chapter 7 Hamming Code
Bit Position 1 2 3 4 5 6 7 8 9 10 11 12 13 Notation P1 P2 B3 P4 B5 B6 B7 P8 B9 B10 B11 B12 P Data 1 1 0 0 0 1 0 0 Transmitted data P1 Xor of bits (3, 5, 7, 9, 11) P2 Xor of bits (3, 6, 7, 10, 11) P4 Xor of bits (5, 6, 7, 12) P8 Xor of bits (9, 10, 11, 12) P Even or Odd parity on bit 112

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Text Book Digital Design 4th Ed.
Chapter 7 Hamming Code
Bit Position 1 2 3 4 5 6 7 8 9 10 11 12 13 Notation P1 P2 B3 P4 B5 B6 B7 P8 B9 B10 B11 B12 P Data 1 1 0 0 0 1 0 0 Received data C1 Xor of bits (1, 3, 5, 7, 9, 11) C2 Xor of bits (2, 3, 6, 7, 10, 11) C4 Xor of bits (4, 5, 6, 7, 12) C8 Xor of bits (8, 9, 10, 11, 12) P Even or Odd parity on bit 112 C 0, P 0 no error occurred C ! 0, P 1 single error occurred, can be corrected C ! 0, P 0 double error occurred, cannot be corrected C 0, P 1 error occurred in the 13th bit

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Text Book Digital Design 4th Ed.
Chapter 7 Hamming Code
7-10 Given the 8-bit data word 0101 1011, generate the 13-bit composite word for the Hamming code that corrects single errors and detect double errors. Bit Position 1 2 3 4 5 6 7 8 9 10 11 12 13 Notation P1 P2 B3 P4 B5 B6 B7 P8 B9 B10 B11 B12 PE Data 0 1 0 1 1 0 1 1 Parity 0 0 1 1 1 Parity data P1 Xor of bits (3, 5, 7, 9, 11) 0 P2 Xor of bits (3, 6, 7, 10, 11) 0 P4 Xor of bits (5, 6, 7, 12) 1 P8 Xor of bits (9, 10, 11, 12) 1 PE Even or Odd parity on bit 112 1

Fuw-Yi Yang
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Text Book Digital Design 4th Ed.
Chapter 7 Hamming Code
7-11 Obtain the 15-bit Hamming code word for the 11-bit data word 1100 1001 010. Position 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Notation P1 P2 B3 P4 B5 B6 B7 P8 B9 B10 B11 B12 B13 B14 B15 Data 1 1 0 0 1 0 0 1 0 1 0 Parity Parity data P1 Xor of bits (3, 5, 7, 9, 11, 13, 15) 1 P2 Xor of bits (3, 6, 7, 10, 11, 14, 15) 0 P4 Xor of bits (5, 6, 7, 12, 13, 14, 15) 1 P8 Xor of bits (9, 10, 11, 12, 13, 14, 15) 1

Fuw-Yi Yang
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Text Book Digital Design 4th Ed.
Chapter 7 Hamming Code
7-12 A 12-bit Hamming code word containing 8 bits of data and 4 parity bits is read from memory. What was the original 8-bit data word that was written into memory if the 12-bit word read out is as follows (a) 0000 1110 1010 (b) 1011 1000 0110 (c) 1011 1111 0100

Fuw-Yi Yang
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Text Book Digital Design 4th Ed.
Chapter 7 Hamming Code
7-13 How many parity check bits must be included with the data word to achieve single-error correction and double-error detection when the data word contains (a) 16 bits (b) 32 bits (c) 48 bits

Fuw-Yi Yang
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Text Book Digital Design 4th Ed.
Chapter 7 Hamming Code
7-14 It is necessary to formulate the Hamming code for four data bits D3, D5, D6, and D7, together with three parity bits, P1, P2, and P4. (a) Evaluate the 7-bit composite code word for the data word 0010. (b) Evaluate the three check bits, C1, C2, and C4, assuming no error. (c) Assume an error in bit D5during writing into memory. Show how the error in the bit is detected and corrected. (d) Add parity bit P8 to include double-error detection in the code. Assume that errors occurred in bits P2 and D5. Show how the double error is detected.

Fuw-Yi Yang
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