Skewb Ultimate

In coding theory, a parity-check matrix of a linear block code C is a generator matrix of the dual code. As such, a codeword c is in C if and only if the matrix-vector product Hc^t = 0.

The rows of a parity check matrix are parity checks on the codewords of a code. That is, they show how linear combinations of certain digits of each codeword equal zero. For example, the parity check matrix

$H=\left[\begin{array}{cccc}0& 0& 1& 1\\ 1& 1& 0& 0\end{array}\right]$

specifies that for each codeword, digits 1 and 2 should sum to zero (according to the second row) and digits 3 and 4 should sum to zero (according to the first row).

Creating a parity check matrix

The parity check matrix for a given code can be derived from its generator matrix (and vice-versa). If the generator matrix for an [n,k]-code is in standard form

$G=\left[\begin{array}{c}{I}_k|P\end{array}\right]$,

then the parity check matrix is given by

$H=\left[\begin{array}{c}-P^T|I_{n-k}\end{array}\right]$,

because

$G{H}^T=P-P=0$.

Negation is performed in the finite field F_q. Note that if the characteristic of the underlying field is 2 (i.e., 1 + 1 = 0 in that field), as in binary codes, then -P = P, so the negation is unnecessary.

For example, if a binary code has the generator matrix

$G=\left[\begin{array}{ccccc}1& 0& 1& 0& 1\\ 0& 1& 1& 1& 0\end{array}\right],$

then its parity check matrix is

$H=\left[\begin{array}{ccccc}1& 1& 1& 0& 0\\ 0& 1& 0& 1& 0\\ 1& 0& 0& 0& 1\end{array}\right].$

For any (row) vector x of the ambient vector space, s = Hx^t is called the syndrome of x. The vector x is a codeword if and only if s = 0.

See also

• Hamming code

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534