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Paritycheck niatrix

An (/i.A) linear codę can be uniquely defined by a system of m-n-k linear equations that express parity check bits by the information bits. It is normally assumed that weight of each codeword is even. Such codę is called the even-parity błock codę. The parity-check matrix can be introduccd:

Hu *	" "u,
	• Hmn

Then, linear codę is defined by m equations, parity-check equations:

0; / = 1.....m

>=•

where, y© is mod-2 addition and w =    is arbitrary codeword,

or in a matrix form:

~ H \ i •	• Hu			0
;	•	:		:
Hml •	-1 i a:	_^w*.		0

For the systematic codę: w = [i, • • -i_k p_x • • • p_m ],

//„ H_u 1    .. 0

H_al -. H* 0 .. 1.

and parity-check sums:

For an (njc) nonsystematic codę. it is normally assumed that all m check bits are also detcrmincd solely by k information bits. i.c. in each parity-check cquation (5.2.12a) only one check bit is present. That way, check bits can be designatcd directly froni information bits. without solving a system of equations. It means, that columns of H that are in check bit positions should contain only one nonzero entry.