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Nordstrom-Robinson (NR) code[1,2]

Description

A nonlinear \((16,256,6)\) binary code that is the smallest Kerdock and the smallest Preparata code. The size of this code is larger than the largest possible linear code with the same length and distance.

The code can be shortened to produce optimal \((15, 128, 6)\), \((14, 64, 6)\) and \((13, 32, 6)\) codes, as well as unique \((15, 256, 5)\), \((15, 128, 6)\), \((14, 128, 5)\), \((14, 64, 6)\), \((13, 64, 5)\) and \((13, 32, 6)\) codes [3; pg. 74]. Further shortening yields the \((12, 32, 5)\) Nadler code [4].

The automorphism group of the code is \(\mathbb{Z}_2^4 \times A_7\) [57][3; pg. 478].

Cousins

Primary Hierarchy

Parents
The NR code is the smallest Kerdock code.
The NR code is the smallest Preparata code.
Nordstrom-Robinson (NR) code

References

[1]
A. W. Nordstrom and J. P. Robinson, “An optimum nonlinear code”, Information and Control 11, 613 (1967) DOI
[2]
N. V. Semakov, V. A. Zinovev, Complete and Quasi-complete Balanced Codes, Probl. Peredachi Inf., 5:2 (1969), 14–18; Problems Inform. Transmission, 5:2 (1969), 11–13
[3]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[4]
J. van Lint, “A new description of the Nadler code (Corresp.)”, IEEE Transactions on Information Theory 18, 825 (1972) DOI
[5]
E. R. Berlekamp, “Coding theory and the Mathieu groups”, Information and Control 18, 40 (1971) DOI
[6]
J.-M. Goethals, “On the Golay perfect binary code”, Journal of Combinatorial Theory, Series A 11, 178 (1971) DOI
[7]
Snover, S. L. (1973). THE UNIQUENESS OF THE NORDSTROM-ROBINSON AND THE GOLAY BINARY-CODES. Michigan State University.
[8]
Forney Jr GD, Sloane NJ, Trott MD. The Nordstrom-Robinson code is the binary image of the octacode. In Coding and Quantization: DIMACS/IEEE workshop 1992 Oct 19 (pp. 19-26). Amer. Math. Soc..
[9]
Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
[10]
M. Shi, T. Honold, P. Sole, Y. Qiu, R. Wu, and Z. Sepasdar, “The Geometry of Two-Weight Codes Over ℤ\({}_{\text{\textit{p}}}\)\({}^{\text{\textit{m}}}\)”, IEEE Transactions on Information Theory 67, 7769 (2021) DOI
[11]
W. Kantor, “On the inequivalence of generalized Preparata codes”, IEEE Transactions on Information Theory 29, 345 (1983) DOI
[12]
A. R. Calderbank, A. R. Hammons Jr., P. V. Kumar, N. J. A. Sloane, and P. Solé, “A linear construction for certain Kerdock and Preparata codes”, (1993) arXiv:math/9310227
[13]
J.-M. Goethals, “The extended Nadler code is unique (Corresp.)”, IEEE Transactions on Information Theory 23, 132 (1977) DOI
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Zoo Code ID: nordstrom_robinson

Cite as:
“Nordstrom-Robinson (NR) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/nordstrom_robinson
BibTeX:
@incollection{eczoo_nordstrom_robinson, title={Nordstrom-Robinson (NR) code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/nordstrom_robinson} }
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“Nordstrom-Robinson (NR) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/nordstrom_robinson

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/nonlinear/gray_map/originals/nordstrom_robinson.yml.