## Description

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 \(n\). Extensions to prime-power \(q\) are also known [1,2].

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.

The automorphism group of extended odd-like quadratic residue codes is \(PSL(2,GF(q))\), and these codes are the only codes with such symmetries [3].

## Rate

## Notes

## Parent

- \(q\)-ary duadic code — QR codes are duadic codes of prime length satisfying certain relations [7].

## Children

- Binary quadratic-residue (QR) code
- Hexacode — The hexacode is the smallest example of an extended quadratic residue code of Type \(4^H\) [8; Sec. 2.4.6][6; Exer. 363]. The shortened hexacode is an odd-like quadratic residue code [6; Ex. 6.6.8].
- Ternary Golay code — The ternary Golay code is a quadratic residue code over \(GF(3)\) with residue set \(Q = \{1, 3, 4, 5, 9\} \) and generator polynomial \(x^5 + x^4 - x^3 + x^2 - 1\) ([5], Ch. 16).

## Cousins

- Combinatorial design — The supports of fixed-weight codewords of certain \(q\)-ary QR codes support combinatorial designs [9–11].
- Quantum quadratic-residue (QR) code

## References

- [1]
- J. van Lint and F. MacWilliams, “Generalized quadratic residue codes”, IEEE Transactions on Information Theory 24, 730 (1978) DOI
- [2]
- J. H. Lint, “Generalized quadratic-residue codes”, Algebraic Coding Theory and Applications 285 (1979) DOI
- [3]
- C. Ding, H. Liu, and V. D. Tonchev, “All binary linear codes that are invariant under \(\PSL_2(n)\)”, (2017) arXiv:1704.01199
- [4]
- K. Ivanov and R. L. Urbanke, “Capacity-achieving codes: a review on double transitivity”, (2020) arXiv:2010.15453
- [5]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
- [6]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [7]
- V. Pless, “Duadic Codes and Generalizations”, Eurocode ’92 3 (1993) DOI
- [8]
- Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
- [9]
- E. F. Assmus, Jr. and H. F. Mattson, Jr., “Coding and Combinatorics”, SIAM Review 16, 349 (1974) DOI
- [10]
- E. F. Assmus Jr. and H. F. Mattson Jr., “New 5-designs”, Journal of Combinatorial Theory 6, 122 (1969) DOI
- [11]
- M. HUBER, “CODING THEORY AND ALGEBRAIC COMBINATORICS”, Selected Topics in Information and Coding Theory 121 (2010) arXiv:0811.1254 DOI

## Page edit log

- Victor V. Albert (2022-07-15) — most recent

## Cite as:

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