A quadratic residue code is a type of cyclic code.

Examples

Examples of quadratic residue codes include the (7,4) Hamming code over GF(2), the (23,12) binary Golay code over GF(2) and the (11,6) ternary Golay code over GF(3).

Constructions

There is a quadratic residue code of length p over the finite field GF(l) whenever p and l are primes, p is odd, and l is a quadratic residue modulo p. Its generator polynomial as a cyclic code is given by where Q is the set of quadratic residues of p in the set and \zeta is a primitive pth root of unity in some finite extension field of GF(l). The condition that l is a quadratic residue of p ensures that the coefficients of f lie in GF(l). The dimension of the code is (p+1)/2. Replacing \zeta by another primitive p-th root of unity \zeta^r either results in the same code or an equivalent code, according to whether or not r is a quadratic residue of p. An alternative construction avoids roots of unity. Define for a suitable c\in GF(l). When l=2 choose c to ensure that g(1)=1. If l is odd, choose , where p^*=p or -p according to whether p is congruent to 1 or 3 modulo 4. Then g(x) also generates a quadratic residue code; more precisely the ideal of generated by g(x) corresponds to the quadratic residue code.

Weight

The minimum weight of a quadratic residue code of length p is greater than \sqrt{p}; this is the square root bound.

Extended code

Adding an overall parity-check digit to a quadratic residue code gives an extended quadratic residue code. When p\equiv 3 (mod 4) an extended quadratic residue code is self-dual; otherwise it is equivalent but not equal to its dual. By the Gleason–Prange theorem (named for Andrew Gleason and Eugene Prange), the automorphism group of an extended quadratic residue code has a subgroup which is isomorphic to either PSL_2(p) or SL_2(p).

Decoding Method

Since late 1980, there are many algebraic decoding algorithms were developed for correcting errors on quadratic residue codes. These algorithms can achieve the (true) error-correcting capacity of the quadratic residue codes with the code length up to 113. However, decoding of long binary quadratic residue codes and non-binary quadratic residue codes continue to be a challenge. Currently, decoding quadratic residue codes is still an active research area in the theory of error-correcting code.