TY - JOUR

T1 - On the existence of tight relative 2-designs on binary Hamming association schemes

AU - Bannai, Eiichi

AU - Bannai, Etsuko

AU - Bannai, Hideo

N1 - Funding Information:
Eiichi Bannai was supported in part by NSFC grant No. 11271257 . Hideo Bannai was supported in part by Kakenhi No. 22680013 and No. 25280086 .

PY - 2014

Y1 - 2014

N2 - It is known that there is a close analogy between "Euclidean t-designs vs. spherical t-designs" and "Relative t-designs in binary Hamming association schemes vs. combinatorial t-designs". In this paper, we want to prove how much we can develop a similar theory in the latter situation, imitating the theory in the former one. We first prove that the weight function is constant on each shell for tight relative t-designs on p shells on a wide class of Q-polynomial association schemes, including Hamming association schemes. In the theory of Euclidean t-designs on 2 concentric spheres (shells), it is known that the structure of coherent configurations is naturally attached. However, it seems difficult to prove this claim in a general context. In the case of tight 2-designs in combinatorial 2-designs, there are great many tight 2-designs, i.e., symmetric 2-designs, while there are very few tight 2e-designs for e≥2. So, as a starting point, we concentrate our study to the existence problem of tight relative 2-designs, in particular on 2 shells, in binary Hamming association schemes H(n,2). We prove that every tight relative 2-design on 2 shells in H(n,2) has the structure of coherent configuration. We determine all the possible parameters of coherent configurations attached to such tight relative 2-designs for n≤30. Moreover for each of them we determine whether there exists such a tight relative 2-design or not, either by constructing them from symmetric 2-designs or Hadamard matrices, or theoretically showing the non-existence. In particular, we show that for n≡6(mod8), there exist such tight relative 2-designs whose weight functions are not constant. These are the first examples of those with non-constant weight.

AB - It is known that there is a close analogy between "Euclidean t-designs vs. spherical t-designs" and "Relative t-designs in binary Hamming association schemes vs. combinatorial t-designs". In this paper, we want to prove how much we can develop a similar theory in the latter situation, imitating the theory in the former one. We first prove that the weight function is constant on each shell for tight relative t-designs on p shells on a wide class of Q-polynomial association schemes, including Hamming association schemes. In the theory of Euclidean t-designs on 2 concentric spheres (shells), it is known that the structure of coherent configurations is naturally attached. However, it seems difficult to prove this claim in a general context. In the case of tight 2-designs in combinatorial 2-designs, there are great many tight 2-designs, i.e., symmetric 2-designs, while there are very few tight 2e-designs for e≥2. So, as a starting point, we concentrate our study to the existence problem of tight relative 2-designs, in particular on 2 shells, in binary Hamming association schemes H(n,2). We prove that every tight relative 2-design on 2 shells in H(n,2) has the structure of coherent configuration. We determine all the possible parameters of coherent configurations attached to such tight relative 2-designs for n≤30. Moreover for each of them we determine whether there exists such a tight relative 2-design or not, either by constructing them from symmetric 2-designs or Hadamard matrices, or theoretically showing the non-existence. In particular, we show that for n≡6(mod8), there exist such tight relative 2-designs whose weight functions are not constant. These are the first examples of those with non-constant weight.

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U2 - 10.1016/j.disc.2013.09.013

DO - 10.1016/j.disc.2013.09.013

M3 - Article

AN - SCOPUS:84885619958

SN - 0012-365X

VL - 314

SP - 17

EP - 37

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 1

ER -