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Hamming Code

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**Error Detecting and Correcting Code**

Parity code Detect odd number of errors Cannot correct any errors Hamming Code Detect up to two simultaneous bit errors Correct single-bit errors

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**Hamming Code For each integer m > 2 Basically**

Code exists with m parity bits and 2m – m – 1 Basically Parity bit for odd bits Parity bit for each two bits Parity bit for each four bits Parity bit for each eight bits Parity bit for each group of bits that are power of 2 1, 2, 4, 8, 16, 32, …

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**Calculating Hamming Code**

Mark all bit positions that are powers of two parity bit 1, 2, 4, 8, 16, 32, 64, etc All other bit positions are for the data to be encoded 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, etc. Each parity bit calculates the parity for some of the bits in the code word The position of the parity bit determines the sequence of bits that it alternately checks and skips Position 1: check 1 bit, skip 1 bit, check 1 bit, skip 1 bit, etc. (1,3,5,7,9,11,13,15,...) Position 2: check 2 bits, skip 2 bits, check 2 bits, skip 2 bits, etc. (2,3,6,7,10,11,14,15,...) Position 4: check 4 bits, skip 4 bits, check 4 bits, skip 4 bits, etc. (4,5,6,7,12,13,14,15,20,21,22,23,...) Position 8: check 8 bits, skip 8 bits, check 8 bits, skip 8 bits, etc. (8-15,24-31,40-47,...) Set a parity bit to 1 if the total number of ones in the positions it checks is odd. Set a parity bit to 0 if the total number of ones in the positions it checks is even

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Hamming Code Layout 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Pos 20=1 21=2 22=4 23=8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Even parity calculated for each bit position P1 (1, 3, 5, 7, 9, 11, 13, 15) P2 (2, 3, 6, 7, 10, 11, 14, 15) P4 (4, 5, 6, 7, 12, 13, 14, 15) P8(8, 9, 10, 11, 12, 13, 14, 15)

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**Hamming Code Layout 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 P1 P2 P4 P8**

P1 P2 P4 P8 P1 = 1 P2 = 1 P4 = 0 P8 = 1 For code: Even parity calculated for each bit position P1 (1, 3, 5, 7, 9, 11, 13, 15) P2 (2, 3, 6, 7, 10, 11, 14, 15) P4 (4, 5, 6, 7, 12, 13, 14, 15) P8(8, 9, 10, 11, 12, 13, 14, 15)

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**Hamming Code (1 error) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 P1 P2 P4 P8**

P1 P2 P4 P8 Xmit Rec A S P1 = 1 1 P2 = 1 0 P4 = 0 1 P8 = 1 0 A = Actual Parity S = Should be Parity Even parity calculated for each bit position P1 (1, 3, 5, 7, 9, 11, 13, 15) P2 (2, 3, 6, 7, 10, 11, 14, 15) P4 (4, 5, 6, 7, 12, 13, 14, 15) P8(8, 9, 10, 11, 12, 13, 14, 15) 2, 4, and 8 incorrect = 14

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**Hamming Code (2 errors) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 P1 P2 P4**

P1 P2 P4 P8 Xmit Rec A S P1 = 1 0 P2 = 1 0 P4 = 0 1 P8 = 1 1 A = Actual Parity S = Should be Parity Even parity calculated for each bit position P1 (1, 3, 5, 7, 9, 11, 13, 15) P2 (2, 3, 6, 7, 10, 11, 14, 15) P4 (4, 5, 6, 7, 12, 13, 14, 15) P8(8, 9, 10, 11, 12, 13, 14, 15) 1,2, and 4 incorrect = 7 Fix 7, will still have parity problem

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Example A byte of data: Create the data word, leaving spaces for the parity bits: _ _ 1 _ _ Calculate the parity for each parity bit (a ? represents the bit position being set): Position 1 checks bits 1,3,5,7,9,11: ? _ 1 _ _ Even parity so set position 1 to a 0: 0 _ 1 _ _ Position 2 checks bits 2,3,6,7,10,11: 0 ? 1 _ _ Odd parity so set position 2 to a 1: 0 1 1 _ _ Position 4 checks bits 4,5,6,7,12: 0 1 1 ? _ Odd parity so set position 4 to a 1: _ Position 8 checks bits 8,9,10,11,12: ? Even parity so set position 8 to a 0: Code word:

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**Finding and Fixing a Bad Bit**

Transmitted word: Received word: Calculate the parity bits from received word SB A P1 (1, 3, 5, 7, 9, 11) = P2 (2, 3, 6, 7, 10, 11) = P4 (4, 5, 6, 7, 12) = P8 (8, 9, 10, 11, 12) = 2 + 8 = 10 Bit position 10 is incorrect

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**Finding and Fixing a Bad Bit**

Transmitted word: Received word: Calculate the parity bits from received word SB A P1 (1, 3, 5, 7, 9, 11) = 0 0 P2 (2, 3, 6, 7, 10, 11) = 0 1 P4 (4, 5, 6, 7, 12) = 0 1 P8 (8, 9, 10, 11, 12) = 0 0 Bit positions 2 & 4 are incorrect 2 + 4 = 6 Correct code is

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**Finding and Fixing a Bad Bit**

Transmitted word: Received word: Calculate the parity bits from received word SB A P1 (1, 3, 5, 7, 9, 11) = 0 1 P2 (2, 3, 6, 7, 10, 11) = 1 1 P4 (4, 5, 6, 7, 12) = 0 1 P8 (8, 9, 10, 11, 12) = 0 0 Bit positions 1 & 4 are incorrect 1 + 4 = 5 Correct code is

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**Finding and Fixing a Bad Bit**

Transmitted word: Received word: Calculate the parity bits from received word SB A P1 (1, 3, 5, 7, 9, 11) = 1 0 P2 (2, 3, 6, 7, 10, 11) = 1 0 P4 (4, 5, 6, 7, 12) = 1 0 P8 (8, 9, 10, 11, 12) = 0 0 Bit positions 1, 2 & 4 are incorrect = 7 Correct code is:

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