Abstract
The notion of T-design in a symmetric association scheme was introduced by Delsarte. A harmonic index t-design is the particular case in which the index set T consists of a single index \(\{t\}\). Zhu et al. studied a harmonic index t-design in the binary Hamming scheme and gave a Fisher type lower bound on the cardinality. Also they defined the notion of a tight harmonic index design using the bound, and considered the classification problem. In this paper, we extend their results to the general Hamming scheme and improve their Fisher type lower bound slightly. Also using the improved Fisher type lower bound, we redefine the notion of a tight harmonic index design and consider the classification problem. Furthermore, we give a natural characterization for a harmonic index t-design and an analogue in the Hamming scheme for the construction of spherical harmonic index t-designs which was given by Bannai–Okuda–Tagami.
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08 July 2024
A Correction to this paper has been published: https://doi.org/10.1007/s00373-024-02810-8
References
Bannai, E., Bannai E., Ito, T: Introduction to Algebraic Combinatorics. Kyoritsu (2016) (in Japanese)
Bannai, E., Bannai, E., Tanaka, H., Zhu, Y.: Design theory from the viewpoint of algebraic combinatorics. Graphs Combin. 33(1), 1–41 (2017)
Bannai, E., Ito, T.: Algebraic Combinatorics I. Association Schemes. Benjamin/Cummings, Menlo Park (1984)
Bannai, E., Okuda, T., Tagami, M.: Spherical designs of harmonic index $t$. J. Approx. Theory 195, 1–18 (2015)
Bogdanova, G., Todorov, T., Yorgova, T.: New ternary and quaternary constant-weight equidistant codes. Discrete Math. Algorithms Appl. 2(1), 89–97 (2010)
Brouwer, A.E., Cioabă, S.M., Ihringer, F., McGinnis, M.: The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters. J. Combin. Theory Ser. B 133, 88–121 (2018)
Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, New York (1989)
Brouwer, A.E., Haemers, W.H.: Association schemes. In: Handbook of Combinatorics, vol. 1. Elsevier, Amsterdam (1995)
Delsarte, P.: An algebraic approach to the association schemes of coding theory, Thesis, Universite Catholique de Louvain. Philips Res. Rep. Suppl., vol. 10 (1973)
Delsarte, P.: Association schemes and $t$-designs in regular semilattices. J. Combin. Theory (A) 20, 230–243 (1976)
Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata 6, 363–388 (1977)
Delsarte, P., Levenshtein, V.I.: Association schemes and coding theory. IEEE Trans. Inf. Theory 44(6), 2477–2504 (1998)
Hall Jr, M.: Combinatorial Theory, Reprint of the 1986, 2nd edn. A Wiley-Interscience Publication, Wiley Classics Library (1998)
Hori, R.: On HI t-designs in Hamming scheme, Master’s thesis, Kyushu Institute of Technology (2016) (in Japanese)
Isaacs, I.M.: Character Theory of Finite Groups. Academic Press, New York (1976)
Stanton, D.: $t$-designs in classical association schemes. Graphs Combin. 2(3), 283–286 (1986)
Terras, A.: Fourier Analysis on Finite Groups and Applications. London Mathematical Society, London (1999)
Zhu, Y., Bannai, E., Bannai, E., Ikuta, T., Kim, K.: Harmonic index designs in binary hamming schemes. Graphs Combin. 33(3), 1–14 (2017)
Acknowledgements
We are grateful to the anonymous referee for the much useful and helpful comment to the paper. M. Tagami is supported by JSPS KAKENHI Grant Number JP 19K03425.
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Dedicated to Professor Eiichi Bannai on the occasion of his 75-th birthday.
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This paper is a reprint of Tagami M. & Hori R., Harmonic Index t-Designs in the Hamming Scheme for Arbitrary q, Graphs Combin. 36(4), 1093–1109 (2020), doi: https://doi.org/10.1007/s00373-020-02170-z. It was submitted to this Special Issue dedicated to Professors Eiichi Bannai and Hikoe Enomoto on their 75th birthdays, but by mistake appeared in the aforementioned volume and issue. The Publisher apologizes for the mistake made and any inconvenience caused.
Appendix
Appendix
The following is the table of \(B_{n,t}\)’s for \(3\le t \le 5\) in which \(B_{n,t}\) is an integer, and in which the condition of Corollary 1.1 holds when q is a prime power.
t | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
q | 2 | 3 | 4 | 4 | 4 | 5 | 6 | 2 | 2 | 2 | 2 | 3 |
n | 4–10 | 4–10 | 4 | 5 | 7–10 | 9–10 | 10 | 6 | 7–8 | 9 | 10 | 5 |
\(B_{n,t}\) | 2 | 9 | 10 | 16 | 28 | 65 | 126 | 4 | 8 | 10 | 16 | 6 |
t | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
|---|---|---|---|---|---|---|---|---|---|---|---|
q | 3 | 4 | 5 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | 6 |
n | 7 | 5–6 | 6 | 6–10 | 7–8 | 10 | 6 | 10 | 6–7 | 6 | 7 |
\(B_{n,t}\) | 15 | 16 | 25 | 2 | 15 | 33 | 10 | 82 | 25 | 26 | 36 |
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Tagami, M., Hori, R. Harmonic Index t-Designs in the Hamming Scheme for Arbitrary q. Graphs and Combinatorics 37, 1669–1685 (2021). https://doi.org/10.1007/s00373-021-02397-4
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DOI: https://doi.org/10.1007/s00373-021-02397-4