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 \(\mathbb{Z}_2^4 \times A_7\) [5–7][3; pg. 478].
Cousins
- Octacode— The NR code is the image of the octacode under the Gray map [8,9]. The \((14, 64, 6)\) shortened NR code is the image of the heptacode under the Gray map [10; Exam. 5].
- \([24, 12, 8]\) Extended Golay code— The NR code can be constructed using the extended Golay code by first selecting a set of codewords satisfying certain conditions and then deleting 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 [11]. It maps to the octacode, a self-dual code over \(\mathbb{Z}_4\) under the Gray map [8,9,12].
- Combinatorial design— 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].
- \((12,32,5)\) Nadler code— The NR code is an extension of the Nadler code [4,13].
Primary Hierarchy
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
Page edit log
- Victor V. Albert (2024-01-11) — most recent
- Madhura Pankaja (2024-01-11)
Cite as:
“Nordstrom-Robinson (NR) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/nordstrom_robinson