Thank you for your help I need a solution of this problem AS

Thank you for your help. I need a solution of this problem ASAP. (AMS351 Book : Numbers, Groups & Codes six edition by Hemphreys & Prest)

4. Give the complete coset decoding table for the code given by the generator matrix 0 0 1 0 1 1

Solution

Generator Matrix:

A generator matrix is a matrix whose rows form a basis for a linear code. The code words are all of the linear combinations of the rows of this matrix, that is, the linear code is the row space of its generator matrix.

Generator matrix G of C is a matrix whose rows generate all the elements of C, i.e., if G=(g1,g2,g3,...gk), then every codeword w of C can be represented as

w=c1g1+c2g2+c3g3+.........+ckgk)= cG

in a unique way, where c= (c1,c2,c3,.....ck)

Coset decoding table for generator matrix:

1 0 01 1 0

0 1 0 1 0 1

0 0 1 0 1 1

Parity check matrix:

0 1 1 0 0 1

1 0 1 0 1 0

1 1 0 1 0 0

Syndrome

Coset

000

0

001

e6

010

e5

011

e1

100

e4

101

e2

110

e3

111

e1+e4, e2+e5, e3+e6

v = c + e, where c is the codeword transmitted, and e is the error vector.

Syndrome	Coset
000	0
001	e6
010	e5
011	e1
100	e4
101	e2
110	e3
111	e1+e4, e2+e5, e3+e6