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