\(q\)-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

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
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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.