\(q\)-ary quadratic-residue (QR) code

Member of a quadruple of cyclic \(q\)-ary codes of prime length \(n\) where \(q\) is prime and a quadratic residue modulo \(n\). The codes are constructed using quadratic residues and nonresidues of \(n\).

The roots of the generator polynomial \(r(x)\) of the first code (see Cyclic-to-polynomial correspondence) are all of the inequivalent quadratic residues of \(n\), and the second code's generator polynomial is \((x-1)r(x)\). The roots of the generator polynomial \(a(x)\) of the third code are all inequivalent nonresidues of \(n\), and the fourth code's generator polynomial is \((x-1)a(x)\). The codes corresponding to polynomials \(r,a\) are often called augmented quadratic-residue codes, while the remaining codes are called expurgated.