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

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 \(n\).

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

Introduction of quadratic-residue codes in Refs. [1][2].

Parent

Children

  • Hexacode — The hexacode is the smallest example of an extended quadratic residue code of Type \(4^H\) ([4], Sec. 2.4.6).
  • 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 \(x^5 + x^4 - x^3 + x^2 - 1\) ([1], Ch. 16).

Cousin

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]
Self-dual Codes and Invariant Theory (Springer-Verlag, 2006). DOI

Zoo code information

Internal code ID: q-ary_quad_residue

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Zoo Code ID: q-ary_quad_residue

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
BibTeX:
@incollection{eczoo_q-ary_quad_residue, title={\(q\)-ary quadratic-residue (QR) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/q-ary_quad_residue} }
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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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/cyclic/q-ary_quad_residue.yml.