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Dual Code

The dual code C^_|_ of a linear code C subset= F_q^n over a finite field F_q is its orthogonal complement with respect to the dot product,

C^_|_={y in F_q^n:x·y=0 for every x in C}.

If C has dimension k, then C^_|_ has dimension n-k and (C^_|_)^_|_=C. The transpose of a generator matrix for C^_|_ is a parity check matrix for C. A linear code satisfying C=C^_|_ is called self-dual (MacWilliams and Sloane 1977).

For example, over the finite field with two elements, the linear code C={(0,0),(1,1)} is its own dual code. Orthogonality here is calculated in the finite field, so (1,1)·(1,1)=0.

See also

Generator Matrix, Linear Code, Orthogonal Complement, Parity Check Matrix

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References

MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error-Correcting Codes. Amsterdam, Netherlands: North-Holland, 1977.

Cite this as:

Weisstein, Eric W. "Dual Code." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DualCode.html

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