## 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.

## Parents

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

## 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

## Page edit log

- Victor V. Albert (2023-03-05) — most recent
- Alexander Barg (2023-03-05)
- Victor V. Albert (2023-02-24)

## Cite as:

“Sharp configuration”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/delsarte_optimal