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].

The automorphism group of the code includes the altenating group \(A_7\) [46].

Parents

Cousins

  • Octacode — The NR code is the image of the octacode under the Gray map [7].
  • Golay code — The NR code can be constructed using the extended Golay code by first selecting a set of Golay codewords satisfying certain conditions and then deleteing specific coordinates [3; pg. 73].
  • Self-dual linear code — The NR code is self-dual in that its distance distribution is invariant under the MacWilliams transform [8]. It maps to the octacode, a self-dual code over \(\mathbb{Z}_4\) under the Gray map [9,10].
  • Combinatorial design code — NR codewords give \(3\)-\((16, 6, 4)\), \(3\)-\((16, 8, 3)\), and \(3\)-\((16, 10, 24)\) designs [3; pg. 164].
  • Small-distance block code — The NR code can be shortened to produce unique \((15, 256, 5)\), \((14, 128, 5)\), and \((13, 64, 5)\) codes [3; pg. 74].

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]
E. R. Berlekamp, “Coding theory and the Mathieu groups”, Information and Control 18, 40 (1971) DOI
[5]
J.-M. Goethals, “On the Golay perfect binary code”, Journal of Combinatorial Theory, Series A 11, 178 (1971) DOI
[6]
Snover, S. L. (1973). THE UNIQUENESS OF THE NORDSTROM-ROBINSON AND THE GOLAY BINARY-CODES. Michigan State University.
[7]
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..
[8]
W. Kantor, “On the inequivalence of generalized Preparata codes”, IEEE Transactions on Information Theory 29, 345 (1983) DOI
[9]
A. R. Calderbank et al., “A linear construction for certain Kerdock and Preparata codes”, (1993) arXiv:math/9310227
[10]
Forney Jr GD, Sloane NJ, Trott MD. The Nordstrom-Robinson code is the binary image of the octacode. InCoding and Quantization: DIMACS/IEEE workshop 1992 Oct 19 (pp. 19-26). Amer. Math. Soc..
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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.