Member of a quadruple of cyclic binary codes of prime length \(n=8m\pm 1\) for \(m\geq 1\) 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.
- Group-algebra code — The self-dual \([48,24,12]\) extended quadratic residue code is a group-algebra code [5; Ex. 16.5.1].
- Divisible code — Extended binary quadratic residue codes of length \(8m\) are self-dual doubly-even codes [6; pg. 82].
- \((u|u+v)\)-construction code — The \((u|u+v)\) construction can be used to obtain nonlinear binary quadratic residue codes .
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“Binary quadratic-residue (QR) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/binary_quad_residue