A quadratic residue code is a type of cyclic code.
There is a quadratic residue code of length over the finite field whenever and are primes, is odd and is a quadratic residue modulo . Its generator polynomial as a cyclic code is given by
where is the set of quadratic residues of in the set and is a primitive th root of unity in some finite extension field of . The condition that is a quadratic residue of ensures that the coefficients of lie in . The dimension of the code is
Replacing by another primitive -th root of unity either results in the same code or an equivalent code, according to whether or not is a quadratic residue of .
An alternative construction avoids roots of unity. Define
for a suitable . When choose to ensure that while if is odd where or according to whether is congruent to or modulo . Then also generates a quadratic residue code; more precisely the ideal of generated by corresponds to the quadratic residue code.
The minimum weight of a quadratic residue code of length is greater than ; this is the square root bound.
Adding an overall parity-check digit to a quadratic residue code gives an extended quadratic residue code. When (mod ) an extended quadratic residue code is self-dual; otherwise it is equivalent but not equal to its dual. By a theorem of Gleason and Prange, the automorphism group of an extended quadratic residue code has a subgroup which is isomorphic to either or .
Examples of quadratic residue codes include the Hamming code over, the binary Golay code over and the ternary Golay code over .
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