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Important Linear Block Codes
EELE 6338Dr. Musbah Shaat 01/10/2012
OutlineOutline
• Hamming Codes.• SEC‐DED Codes.SEC DED Codes.• Reed‐Muller Codes.• The (24,12) Golay Code.
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Hamming Codes
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S t ti H i C dSystematic Hamming Codes• In systematic form:
H =[ Im Q]
• The columns of Q are all m-tuple of weight 2.• Different arrangements of the columns of Q produce different• Different arrangements of the columns of Q produce different
codes, but of the same distance property.• Hamming codes are perfect codes
t
P f t d h th t d d t i ll th
t
i
kn
in
0 2
• Perfect code: when the standard array contains all the error pattern of t but no others.
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Weight Distribution of Hamming g gCodes
1 2/)1(2 ))(1()1(1
1)(
nn zznzn
zA
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SEC DED CodesSEC‐DED Codes
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Reed Muller (RM) CodesReed‐Muller (RM) Codes
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Codeword structure
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Example 4 2Example 4.2
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Decoding Example (a start )Decoding Example (a start …)
‐ How these equations are formed ????
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Majority logic Decision RuleMajority‐logic Decision Rule
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Multi stage Decoding ()Multi‐stage Decoding ()
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Multi stage Decoding ()Multi‐stage Decoding ()
N t f i th difi d t• Next, we form again the modified vector
• Decoding of RM(r,m) code consists of r+1 steps. • We start by the information bits of degree r.• The modified vector is constructed after every ydecoding level.
• Called (r+1)‐step majority decoding. ( ) p j y g
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How to construct the check sums ?How to construct the check‐sums ?
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Example 4 3Example 4.3
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Example 4 3 (Cont )Example 4.3 (Cont.)
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Example 4 3 (Cont )Example 4.3 (Cont.)
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Kronecker product ()Kronecker product ()
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Kronecker product ()Kronecker product ()
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Construction of RM code using Kronecker product
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Example 4 4Example 4.4
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The (24 12) Golay CodeThe (24,12) Golay Code
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The (24 12) Golay Code GenerationThe (24,12) Golay Code Generation
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Golay Code Decoding AlgorithmGolay Code Decoding Algorithm
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Decoding Algorithm ContDecoding Algorithm, Cont.
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Decoding Algorithm ContDecoding Algorithm, Cont.
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Decoding Algorithm ContDecoding Algorithm, Cont.
Th l ith b i d• The decoding algorithm can be summarized as follows
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Example 4 7Example 4.7
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• 3.1 , 3.2 , 3.3 and 4.8 (do only one or two information bits in each decoding level).g )
N l ill h li d• Next lecture: we will cover the cyclic codes.
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