\(q\)-ary sharp configuration

\(q\)-ary sharp configuration[1–3]

Description

A \(q\)-ary code that admits \(m\) different distances between distinct codewords and forms a design of strength \(2m-1\) or greater.

Cousins

Hadamard code— Hadamard codes for \(q=4r\) are sharp configurations [4; Table 12.1].
Hyperoval code— Codes based on hyperovals in \(PG_{2}(q)\) are \(q\)-ary sharp configurations [4; Table 12.1].
Hill projective-cap code— Hill projective-cap codes for \(n=56\) and \(78\) are \(q\)-ary sharp configurations [4; Table 12.1].

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

\(q\)-ary codeECC

Universally optimal \(q\)-ary codeUniversally optimal ECC

Parents

Universally optimal \(q\)-ary code

All \(q\)-ary sharp configurations are universally optimal \(q\)-ary codes [3], but the converse is not true.

Sharp configuration\(t\)-design Universally optimal ECC

Orthogonal array (OA)\(t\)-design ECC

\(q\)-ary sharp configuration

Children

\([23, 12, 7]\) Golay code

The Golay code and two of its shortened versions are \(q\)-ary sharp configurations [4; Table 12.1].

Single parity-check (SPC) code

The SPC code is a binary sharp configuration [4; Table 12.1].

\(q\)-ary repetition code

The \(q\)-ary repetition code is a \(q\)-ary sharp configuration [4; Table 12.1].

Ternary Golay code

The ternary Golay code and one of its shortened versions are \(q\)-ary sharp configurations [4; Table 12.1].

Ovoid code

The ovoid code is a \(q\)-ary sharp configuration [4; Table 12.1].

Denniston code

The Denniston code is a \(q\)-ary sharp configuration [4; Table 12.1].

\(ED_m\) code

The \(ED_m\) code is a \(q\)-ary sharp configuration [4; Table 12.1].

Semakov-Zinoviev-Zaitsev (SZZ) equidistant code

The SZZ equidistant code is a \(q\)-ary sharp configuration [4; Table 12.1].

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 Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
[4]: P. Boyvalenkov, D. Danev, "Linear programming bounds." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) 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:

“\(q\)-ary sharp configuration”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/delsarte_optimal_q-ary

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/universally_optimal/delsarte_optimal_q-ary.yml.