Computing

198 ◾ Computing

For this Hamming code we can decode any received word by the

following set of rules:

1. ere are no errors in the received word if s

1

s

2

s

3

= 000.

2. b

1

is incorrect if s

1

s

2

s

3

= 100.

3. b

2

is incorrect if s

1

s

2

s

3

= 010.

4. b

3

is incorrect if s

1

s

2

s

3

= 001.

5. a

1

is incorrect if s

1

s

2

s

3

= 101.

6. a

2

is incorrect if s

1

s

2

s

3

= 110.

7. a

3

is incorrect if s

1

s

2

s

3

= 111.

8. a

4

is incorrect if s

1

s

2

s

3

= 011.

22.3 LINEAR CODES

Under component-wise vector addition and component-wise scalar multi-

plication, the set of n-tuples of elements from GF(q) is a vector space called

GF(q)

n

, where GF(q) is the nite eld with q elements (also ca ...