1. Hamming Code In 1950s: invented by Richard Hamming
Binary linear cyclic codes
Hamming distance (dmin)=3
Single Error Correction (SEC): detect 1 bit and correct 1 bit
Extended Hamming : (dmin)=4
Single Error Correction- Double Error Detection (SEC-DED): detect double bits and correct single bit
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2. Hamming Code
The Hamming distance is the number of bits that have to be changed to get from one bit pattern to another. The minimum distance is defined as:
dmin= min{d(v, w): v, w ∈C, v ≠ w}
Code length: n - 2r - 1
Number of information symbols: k – 2r – r – 1
Number of parity check symbols: n – k = r
Error correcting capability: t = 1 (dmin = 3)
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3. Hamming Encoding Generator matrix: G= [Ik | P] Codeword, C = i·G.
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4. Parity bit calculation
(7, 4) Hamming code
P1 = XOR bit positions (1, 3, 5, 7)
P2 = XOR bit positions (2, 3, 6, 7)
P3 = XOR bit positions (4, 5, 6, 7)
P1=d1+d2+d4
P2=d1+d3+d4
P3=d2+d3+d4
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5. [7,4] binary Hamming code words
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6. (7,4) Hamming Encoder Fig: Architecture of (7,4) Hamming Encoder
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7. Hamming Decoding Parity check matrix: H= [PT | In-k]
The syndrome, S=v.HT
Table: Syndrome Table
Sl. No.
Error pattern, e
Syndrome (S) 1
000 2
001 3
010 4
100 5
011 6
110 7
111 8
101
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8. Decoding steps Calculate syndrome from the received signal.
From the syndrome table obtain the error pattern that corresponds to particular syndrome.
The required code word is (c)=v+e, this has the effect of inverting the bit in received message given by the position of the non-zero bit in e.
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9. (7,4) Hamming Decoder Fig: Architecture of (7,4) Hamming Decoder
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10. Extended Hamming Code
H-matrix of an extended Hamming Code can be constructed from the check matrix of a Hamming code by adding a zero column on the left and a row of 1’s to the bottom.
H-matrix of (8,4) Hamming code
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11. Application
Hamming Codes are still widely used in computing and telecommunication.
SEC-DED is used in computer memory
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12. Summery
Hamming codes are important channel coding techniques in different digital communication systems and storage system.
Encoding and decoding is very simple.
Hardware Complexity is less.
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13. References
R. W. Hamming, Error detecting and error correcting codes. Bell System technical journal 29(2), , 1950.
Hsiao MY (1970) A class of optimal minimum odd-weight column SEC-DED codes. IBM J. Res. Develop., 14(4):395–401
Gherman V, Evain S, Seymour N, Bonhomme N (2011) Generalized parity-check matrices for SEC-DED codes with fixed parity. Proc. of IEEE On-Line Testing Symp., pp. 198–200
Cha S, Yoon H (2012) Efficient Implementation of Single Error Correction and Double Error Detection Code with Check Bit Pre-computation for Memories. JSTS: Journal of Semiconductor Technology and Science, 12(4):
Reviriego P, Pontarelli S, Maestro JA, Ottavi M (2013) A method to construct low delay single error correction codes for protecting data bits only. IEEE Trans. on Computer-Aided Design of Integ. Circ. and Syst. 32(3):
S. Lin, D. J. Costello, Error Control Coding: Fundamentals and Applications. Prentice-Hall, 1983.
E. Fujiwara, Code Design for Dependable Systems Theory and Practical Applications. John Wiley and Sons Inc., New Jersey, 2006.
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14. THANK YOU
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