Error Control Codes or Channel Codes -Haming Codes

Module IV – Perfect Codes & Hamming Codes
Lakshmi V.S.
Assistant Professor
Electronics & Communication Department
Sree Chitra Thirunal College of Engineering, Trivandrum

Perfect Codes
• Perfect codes are binary codes which satisfies Hamming bound with equality
• A Perfect Code is a t-error correcting code where its standard array has all error patterns of
t or fewer errors and no others as coset leaders
• Examples
 odd-length binary repetition codes (eg. (3, 1), (5,1),..) (these codes contain two
codewords: an all-zero codeword and an all-one codeword),
 binary Hamming codes with dmin = 3
2
Hamming Bound ,2
𝑛 −𝑘
≥∑
𝑗=0
𝑡
(𝑛
𝑗)
Department of Applied Electronics & Instrumentation

Hamming Codes
• Hamming codes are single error correcting perfect codes.
• As t = 1, RHS = =+ =1+n
• LHS = (by considering
• Since Hamming codes satisfies Hamming bound with equality
3

Hamming Codes - Parameters
• Hamming codes are single error correcting perfect codes.
• For any positive integer m≥3, there exists a Hamming code such that:
 Code Length: n = 2m
- 1
 No. of information symbols: k = n-m =2m
- m-1
 No. of parity check symbols: n-k = m
 Error correcting capability: t = 1 (i.e., dmin=3)
• m = 3  (7, 4) Hamming code (n = 2m
-1=7 & k = n-m = 4)
• m = 4  (15, 11) Hamming code (n = 2m
-1=15 & k = n-m = 11)
• m = 5  (31, 26) Hamming code (n = 2m
-1=31 & k = n-m = 26)
4

Hamming Codes
• The parity-check matrix H of Hamming code consists of all possible nonzero m-tuple ((n-k)
tuple) as its columns since n = 2m
- 1
• For (7, 4) systematic Hamming code, H matrix is of size 3 x 7 (n-k x n)
5
PT
3 x 4
I3
3 x 3
I3
3 x 3
PT
3 x 4

Systematic Hamming Codes
•
•
6
I3
3 x 3
PT
3 x 4
P
4 x 3
I4
4 x 4
dmin = 3
edet s  dmin –1, s = 2
ecorr , t = 1

Hamming Codes
• Using the G matrix, the codewords can be generated as v = u.G as in the case of LBC.
• Decoding can be done using standard array and syndrome decoding.
• Encoder circuit, syndrome circuit and decoder circuit are same as that
of LBC.
7

• = .
 v0  u0  u2  u3
 v1  u0  u1  u2
 v2  u1  u2  u3
 v3  u0
 v4  u1
 v5  u2
 v6  u3
8
Encoder – (7, 4) Systematic Hamming Code

Encoder – (7, 4) Systematic Hamming Code
 v0  u0  u2  u3
 v1  u0  u1  u2
 v2  u1  u2  u3
 v3  u0
 v4  u1
 v5  u2
 v6  u3
9

Decoder – (7, 4) Systematic Hamming Code
• = .
 s0  r0  r3  r5  r6
 s1  r1  r3  r4  r5
 s2  r2  r4  r5  r6
10

11

12

13

14


H T =
[
1 0 0
0 1 0
0 0 1
1 1 0
0 1 1
1 1 1
1 0 1
]

Decoding Circuit for (7, 4) Systematic Hamming Code
15
 s0  r0  r3  r5  r6
 s1  r1  r3  r4  r5
 s2  r2  r4  r5  r6



Non Systematic Hamming Codes
• A non-systematic Hamming code can be constructed by placing the parity check bits at 2l
(l
= 0, 1, 2, 3, …) locations of G matrix
• Procedure for constructing non-systematic Hamming code
 Step 1: Write the BCD of length (n – k) for decimals from 1 to n.
 Step 2: Arrange the sequences in the reverse order in a matrix form.
 Step 3: Transpose of the matrix obtained in step 2 gives the parity check matrix, H for
the code.
16

(7, 4) Non Systematic Hamming Codes
 Step 1: Write the BCD of length (n – k) for decimals from 1 to n.
17

 Step 2: Arrange the sequences in the reverse order in a matrix form to form HT
.
 Step 3: Transpose of the matrix obtained in step 2 gives the parity check matrix, H for
the code.
18

• A non-systematic Hamming code can be constructed by placing the parity check bits at 2l
(l
= 0, 1, 2, 3, …) locations of G matrix
• 1, 2 & 4 columns in H matrix are part of identity matrix
• Select each row elements from remaining columns of H as part of PT
and place it as 1, 2 and
4 columns of G.
• Fill the remaining columns of G with columns of identity matrix.
19

• The parity bit equations are
• .The codeword format for non-systematic Hamming code is
20

21
v = u.G

Decoding - (7, 4) Non Systematic Hamming Codes
22
s = e.HT

THANK YOU…
Department of Electronics & Communication 23

Error Control Codes or Channel Codes -Haming Codes

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