# American Institute of Mathematical Sciences

November  2020, 14(4): 603-611. doi: 10.3934/amc.2020033

## Designs from maximal subgroups and conjugacy classes of Ree groups

 1 School of Mathematical Sciences, North-West University, (Mafikeng) 2754, South Africa 2 School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal, Durban 4000, South Africa

* Corresponding author: Seiran Zandi

Received  August 2018 Revised  August 2019 Published  November 2019

Fund Project: The first author acknowledges support of NRF and NWU (Mafikeng).
The second author acknowledges support of NRF through Grant Numbers 95725 and 106071.
The third author acknowledges support of NWU (Mafikeng) postdoctoral fellowship.
The fourth author acknowledges support of NRF postdoctoral fellowship through Grant Number 91495

In this paper, using a method of construction of $1$-designs which are not necessarily symmetric, introduced by Key and Moori in [5], we determine a number of $1$-designs with interesting parameters from the maximal subgroups and the conjugacy classes of the small Ree groups $^2G_2(q)$. The designs we obtain are invariant under the action of the groups $^2G_2(q)$.

Citation: Jamshid Moori, Bernardo G. Rodrigues, Amin Saeidi, Seiran Zandi. Designs from maximal subgroups and conjugacy classes of Ree groups. Advances in Mathematics of Communications, 2020, 14 (4) : 603-611. doi: 10.3934/amc.2020033
##### References:

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##### References:
Non-trivial designs from $G = Ree(q)$ using construction Method 2
 $Max$ $t =o(x)$ $v = |x^G|$ $k =| M \cap x^G|$ $\lambda= \chi_{M_i}(x)$ $M_1$ $t=2$ $q^2(q^2-q+1)$ $q^2$ $q+1$ $M_1$ $t=3$ $(q^3+1)(q-1)$ $q-1$ 1 $M_1$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q(q-1)}{2}$ 1 $M_1$ $t=9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q^2(q-1)}{3}$ 1 $M_1$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q^2 (q-1)}{2}$ 1 $M_1$ $t |(q-1)$, $t \ne 2$ ${q^3}(q^3+1)$ $2q^3$ 2 $M_2, M_3$ $t=2$ $q^2(q^2-q+1)$ $q^{\mp}$ $\frac{{{q(q^2-1)}}}{6}$ $M_2, M_3$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $q^{\mp}$ $\frac{q^2}{3}$ $M_2, M_3$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $q^{\mp}$ $\frac{q}{3}$ $M_2, M_3$ $t | q^{\mp}$ ${q^3(q^2-1)q^{\pm}}$ $6$ $1$ $M_4$ $t=2$ $q^2(q^2-q+1)$ $q^2-q+1$ $q^2-q+1$ $M_4$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q^2-1}{2}$ $q$ $M_4$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q^2-1}{2}$ $1$ $M_4$ $t |(q-1)$, $t \ne 2$ $q^3(q^3+1)$ $q(q+1)$ $1$ $M_4$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $3q(q-1))$ $3$ $M_5$ $t=2$ $q^2(q^2-q+1)$ $q+4$ $\frac{{{q(q-1)(q+4)}}}{6}$ $M_5$ $t=3$ $\frac{{{{q}(q^3+1)(q-1)}}}{2}$ $q+1$ $\frac{{{q^2}}}{3}$ $M_5$ $t=6$ $\frac{{{{q^2}(q^3+1)(q-1)}}}{2}$ $q+1$ $\frac{{{q}}}{3}$ $M_5$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $6$ 1 $M_6$ $t=2$ $q^2(q^2-q+1)$ $q_0^2(q_0^2-q_0+1)$ $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ $M_6$ $t=3$ $(q^3+1)(q-1)$ $(q_0^3+1)(q_0-1)$ $\frac{{{q^3}}}{q_0^3}$ $M_6$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q_0(q_0^3+1)(q_0-1)}{2}$ $\frac{{{q^2}}}{q_0^2}$ $M_6$ $t= 9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{3}$ $\frac{{{q}}}{q_0}$ $M_6$ $t= 6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{2}$ $\frac{{{q}}}{q_0}$ $M_6$ $t |(q_0-1)$, $t \ne 2$ ${q^3(q^3+1)}$ ${q_0^3(q_0^3+1)}$ $\frac{q-1}{q_0-1}$ $M_6$ $t |\frac{q_0+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ ${q_0^3(q_0^2-q_0+1)(q_0-1)}$ $\frac{q+1}{q_0+1}$ $^*M_6$ $t|q_0^{\pm}$ ${q^3}\left( {q^2 - 1} \right){q^{\pm}}$ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}}$ $\frac{{q^{\mp}}}{{q_0^{\mp}}}$ $^{**}M_6$ $t|q_0^{\pm}$ ${q^3(q^3+1)}$ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}}$ $\frac{q-1}{q_0^{\mp}}$
 $Max$ $t =o(x)$ $v = |x^G|$ $k =| M \cap x^G|$ $\lambda= \chi_{M_i}(x)$ $M_1$ $t=2$ $q^2(q^2-q+1)$ $q^2$ $q+1$ $M_1$ $t=3$ $(q^3+1)(q-1)$ $q-1$ 1 $M_1$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q(q-1)}{2}$ 1 $M_1$ $t=9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q^2(q-1)}{3}$ 1 $M_1$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q^2 (q-1)}{2}$ 1 $M_1$ $t |(q-1)$, $t \ne 2$ ${q^3}(q^3+1)$ $2q^3$ 2 $M_2, M_3$ $t=2$ $q^2(q^2-q+1)$ $q^{\mp}$ $\frac{{{q(q^2-1)}}}{6}$ $M_2, M_3$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $q^{\mp}$ $\frac{q^2}{3}$ $M_2, M_3$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $q^{\mp}$ $\frac{q}{3}$ $M_2, M_3$ $t | q^{\mp}$ ${q^3(q^2-1)q^{\pm}}$ $6$ $1$ $M_4$ $t=2$ $q^2(q^2-q+1)$ $q^2-q+1$ $q^2-q+1$ $M_4$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q^2-1}{2}$ $q$ $M_4$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q^2-1}{2}$ $1$ $M_4$ $t |(q-1)$, $t \ne 2$ $q^3(q^3+1)$ $q(q+1)$ $1$ $M_4$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $3q(q-1))$ $3$ $M_5$ $t=2$ $q^2(q^2-q+1)$ $q+4$ $\frac{{{q(q-1)(q+4)}}}{6}$ $M_5$ $t=3$ $\frac{{{{q}(q^3+1)(q-1)}}}{2}$ $q+1$ $\frac{{{q^2}}}{3}$ $M_5$ $t=6$ $\frac{{{{q^2}(q^3+1)(q-1)}}}{2}$ $q+1$ $\frac{{{q}}}{3}$ $M_5$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $6$ 1 $M_6$ $t=2$ $q^2(q^2-q+1)$ $q_0^2(q_0^2-q_0+1)$ $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ $M_6$ $t=3$ $(q^3+1)(q-1)$ $(q_0^3+1)(q_0-1)$ $\frac{{{q^3}}}{q_0^3}$ $M_6$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q_0(q_0^3+1)(q_0-1)}{2}$ $\frac{{{q^2}}}{q_0^2}$ $M_6$ $t= 9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{3}$ $\frac{{{q}}}{q_0}$ $M_6$ $t= 6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{2}$ $\frac{{{q}}}{q_0}$ $M_6$ $t |(q_0-1)$, $t \ne 2$ ${q^3(q^3+1)}$ ${q_0^3(q_0^3+1)}$ $\frac{q-1}{q_0-1}$ $M_6$ $t |\frac{q_0+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ ${q_0^3(q_0^2-q_0+1)(q_0-1)}$ $\frac{q+1}{q_0+1}$ $^*M_6$ $t|q_0^{\pm}$ ${q^3}\left( {q^2 - 1} \right){q^{\pm}}$ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}}$ $\frac{{q^{\mp}}}{{q_0^{\mp}}}$ $^{**}M_6$ $t|q_0^{\pm}$ ${q^3(q^3+1)}$ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}}$ $\frac{q-1}{q_0^{\mp}}$
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