# Binary quadratic-residue (QR) code

## Description

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.

## Notes

## Parent

- Binary duadic code — QR codes are duadic codes of prime length satisfying certain relations [3].

## Child

- Golay code — The Golay code is a binary quadratic residue code with generator polynomial \(r(x)\) over \(GF(2)\) with length \(n=23\) ([1], Ch. 16).

## Cousins

- \(q\)-ary quadratic-residue (QR) code
- Hamming code — \([7,4,3]\) Hamming code is a quadratic-residue code [1].

## References

## Zoo code information

## Cite as:

“Binary quadratic-residue (QR) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/binary_quad_residue