[Jump to code hierarchy]

Sharp configuration[1–3]

Alternative names: Delsarte code.

Description

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 a compact connected two-point homogeneous 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 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.

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].

Member of code lists

Primary Hierarchy

Classical Domain
Error-correcting code (ECC)
Block code
Universally optimal code
Parents
Universally optimal code
All sharp configurations are universally optimal [3,6], but not all universally optimal codes are sharp configurations.
\(t\)-designECC
Sharp configurations attain 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.
Sharp configuration
Children
\(q\)-ary sharp configuration
Spherical sharp configuration

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
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

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} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/delsarte_optimal

Cite as:

“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/edit/main/codes/classical/properties/block/universally_optimal/delsarte_optimal.yml.