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Parity check matrix An ( n , k ) linear code 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 code is called the even-parity block code. The parity-check matrix can be introduced:
Then, linear code is defined by m equations, parity-check equations:
where, is mod-2 addition and is arbitrary codeword, or in a matrix form:
. For the systematic code: ,
and parity-check sums:
For an ( n , k ) nonsystematic code, it is normally assumed that all m check bits are also determined solely by k information bits, i.e. in each parity-check equation (5.2.12a) only one check bit is present. That way, check bits can be designated directly from 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.
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