Harmonic Index Designs in Binary Hamming Schemes

Zhu, Yan; Bannaĭ, Eiichi; Ikuta, Takuya; Kim, Kyoung-Tark

doi:10.1007/s00373-017-1784-5

Cited by 3 publications

(8 citation statements)

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“…; ng, and thus it is always smaller than or equal to one of [18]. Therefore the Fisher type lower bound in this paper improves the bound of [18]. Further, due to their definition of C n;t , they argued only in the case when t is even, but since we consider the minimum of K t ðsÞ only on the finite set (which always exists), one can also argue in the case when t is odd.…”

Section: Now Let Us Give a Fisher Type Lower Bound For An Himentioning

confidence: 63%

“…. ; ng, and thus it is always smaller than or equal to one of [18]. Therefore the Fisher type lower bound in this paper improves the bound of [18].…”

Section: Now Let Us Give a Fisher Type Lower Bound For An Himentioning

confidence: 68%

“…Remark 2 From the proof of Theorem 3, which is given in Sect. A Fisher type lower bound for an HI t-design has been already given by Zhu et al [18] for the binary Hamming schemes, but the bound of Theorem 3 is slightly different from that in [18]. In [18], one corresponding to the above C n;t is defined to be minus times the minimum of K t ðsÞ on the whole of R, which exists only when t is even.…”

Section: Now Let Us Give a Fisher Type Lower Bound For An Himentioning

confidence: 98%

“…A Fisher type lower bound for an HI t-design has been already given by Zhu et al [18] for the binary Hamming schemes, but the bound of Theorem 3 is slightly different from that in [18]. In [18], one corresponding to the above C n;t is defined to be minus times the minimum of K t ðsÞ on the whole of R, which exists only when t is even. While in Theorem 3, C n;t is defined to be minus times the minimum of K t ðsÞ on the finite points set f1; .…”

Section: Now Let Us Give a Fisher Type Lower Bound For An Himentioning

confidence: 98%

“…In that case, we call a ftg-design a harmonic indext-design (or simply an HI t-design) and consider them in the general Hamming scheme. For the case of the binary Hamming scheme, there is the prior research due to Zhu et al [18].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Harmonic Index t-Designs in the Hamming Scheme for Arbitrary q

Tagami

Hori

2021

Graphs and Combinatorics

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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 ftg. 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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Section: Now Let Us Give a Fisher Type Lower Bound For An Himentioning

confidence: 63%

“…. ; ng, and thus it is always smaller than or equal to one of [18]. Therefore the Fisher type lower bound in this paper improves the bound of [18].…”

Section: Now Let Us Give a Fisher Type Lower Bound For An Himentioning

confidence: 68%