66809

Generator niatrix

Any algcbraically independent subset of k nonzero codcwords uniqucly defines an (n,k) linear codę. Such subset consists a code-base, that can be considered as rows of the so called generator matrix:

To obtain all possible combinations of generator matrix rows. matrix G has to be multiplied by a M x k matrix that contains all possible infomiation words. including all-zero word:

*1	0	0 •	• 0	1
i2	0	0 •	• 1	0
	1	I •	• I	1
	0	0 •	• 0	0

2°

2'

2* -1 0

Then.

W = IG

and a single codcword, w = iG.

for simplicity of dcscription. codcword index ; is omitted. Normally. a code-base words are words with W(ij) = I, ordered in such a way that an k *k identity niatrix 1₄ appears at the left side of G:

G

0 Pu - Pu

[l.P]

where. P is called an k*m parity matrix. Such form of G is called the systematie form and the obtaincd codę the systematie codę. In such codę, infomiation bits are located at the left side of a codeword:

Wj=ijJ=\.....M

Codę not having this property is called the nonsystematic codę.