Sharp configuration[1–3] 

Description

Also called a Delsarte code. A code \(C\) that attains a universal bound expressed in terms of the minimal distance, the number of distances between codewords, and the strength of the design formed by the codewords. For codes on the sphere or on the real, complex, quaternionic, or octonionic projective space, \(C\) is a design of strength \(M\) and admits \(m\) different distances between its points such that \(M \geq 2m - 1 - \delta\), where \(\delta\) is one if there are two antipodal points in \(C\) and zero otherwise [3].

Sharp configurations attain the the Levenshtein bound [1,2,4,5]. However, not all codes that attain the Levenshtein bound are sharp configurations. See [1; Table 9.2] for Levenshtein-bound achieving codes on various projective spaces.

Parent

  • Universally optimal code — All sharp configurations are universally optimal [3,6], but not all universally optimal codes are sharp configurations.

Children

Cousins

  • 24-cell code — The 12 sets of antipodal pairs of the 24-cell code form a sharp configuration in the projective space \(\mathbb{R}P^3\) [3].
  • \(E_6\) lattice-shell code — The 36 sets of antipodal pairs of the smallest \(E_6\) lattice shell form a sharp configuration in the projective space \(\mathbb{R}P^5\) [3].
  • \(E_7\) lattice-shell code — The 63 sets of antipodal pairs of the smallest \(E_7\) lattice shell form a sharp configuration in the projective space \(\mathbb{R}P^6\) [3].

References

[1]
V. I. Levenshtein, “Designs as maximum codes in polynomial metric spaces”, Acta Applicandae Mathematicae 29, 1 (1992) DOI
[2]
V. I. Levenshtein, “Universal bounds for codes and designs,” in Handbook of Coding Theory 1, eds. V. S. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998, pp.499-648.
[3]
H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
[4]
V. I. Levenshtein, "On choosing polynomials to obtain bounds in packing problems." Proc. Seventh All-Union Conf. on Coding Theory and Information Transmission, Part II, Moscow, Vilnius. 1978.
[5]
V. I. Levenshtein, “On bounds for packings in n-dimensional Euclidean space”, Dokl. Akad. Nauk SSSR, 245:6 (1979), 1299–1303
[6]
H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
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Zoo Code ID: delsarte_optimal

Cite as:
“Sharp configuration”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/delsarte_optimal
BibTeX:
@incollection{eczoo_delsarte_optimal,
  title={Sharp configuration},
  booktitle={The Error Correction Zoo},
  year={2023},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/delsarte_optimal}
}
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“Sharp configuration”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/delsarte_optimal

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/properties/block/universally_optimal/delsarte_optimal.yml.