## 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

## Parents

- Binary duadic code — QR codes are duadic codes of prime length satisfying certain relations [3].
- \(q\)-ary quadratic-residue (QR) code

## Children

- 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).
- \([48,24,12]\) self-dual code
- \([7,4,3]\) Hamming code — \([7,4,3]\) Hamming code is a quadratic-residue code with generator polynomial \(1+x+x^3\) [1].

## Cousins

- Group-algebra code — The self-dual \([48,24,12]\) extended quadratic residue code is a group-algebra code [4][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 [7].

## References

- [1]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
- [2]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [3]
- V. Pless, “Duadic Codes and Generalizations”, Eurocode ’92 3 (1993) DOI
- [4]
- I. McLoughlin, “A group ring construction of the [48,24,12] type II linear block code”, Designs, Codes and Cryptography 63, 29 (2011) DOI
- [5]
- W. Willems, "Codes in Group Algebras." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [6]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [7]
- N. Sloane and D. Whitehead, “New family of single-error correcting codes”, IEEE Transactions on Information Theory 16, 717 (1970) DOI

## Page edit log

- Victor V. Albert (2022-07-15) — most recent
- Yijia Xu (2022-04-25)

## 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