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

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

Parents

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

Cousins

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
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Zoo Code ID: binary_quad_residue

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/bits/cyclic/binary_quad_residue.yml.