A parity check matrix 
of a linear code 
of length 
and dimension 
over a field 
is an 
matrix whose columns
form a basis for the dual
code 
.
A row vector 
is a codeword of 
iff 
. Equivalently, the columns of 
form a basis for the null
space of a generator matrix 
, and 
is a generator matrix
for the dual code. A convention using the transpose
of 
instead writes the check as 
.
Parity Check Matrix
See also
Coding Theory, Dual Code, Error-Correcting Code, Generator Matrix, Linear CodePortions of this entry contributed by David Terr
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References
Roman, S. Coding and Information Theory. New York: Springer-Verlag, 1992.van Lint, J. H. Introduction to Coding Theory, 2nd ed. Berlin, Germany: Springer-Verlag, 1992. https://doi.org/10.1007/978-3-662-00174-5.Referenced on Wolfram|Alpha
Parity Check MatrixCite this as:
Weisstein, Eric W., with contributions by David Terr. "Parity Check Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ParityCheckMatrix.html