#!/usr/bin/env python
# coding: utf-8

# # Nonexistence of a $Q$-polynomial association scheme with Krein array $\{24, 20, 36/11; 1, 30/11, 24\}$
# 
# We will show that a $Q$-polynomial association scheme with Krein array $\{24, 20, 36/11; 1, 30/11, 24\}$ does not exist.

# In[1]:


get_ipython().run_line_magic('display', 'latex')
import drg


# Such a scheme would have $225$ vertices.

# In[2]:


p = drg.QPolyParameters([24, 20, 36/11], [1, 30/11, 24])
p.order()


# The scheme has two $Q$-polynomial orderings $(0, 1, 2, 3)$ and $(0, 3, 2, 1)$, so we have $q^3_{11} = q^1_{13} = q^1_{31} = q^1_{33} = q^3_{13} = q^3_{31} = 0$.

# In[3]:


p.kreinParameters()


# The intersection numbers are integral and nonnegative.

# In[4]:


p.pTable()


# The parameters do not exceed the absolute bound.

# In[5]:


p.check_absoluteBound()


# Let $w, x, y, z$ be vertices such that $z$ is in relation $1$ with $x$ and $y$, and $w, x, y$ are mutually in relation $3$. Note that we have $p^3_{11} = 22$ and $p^3_{33} = 3$, so such vertices must exist. We first compute the triple intersection numbers with respect to $x, y, z$. The parameters $\alpha$ and $\beta$ will denote the numbers of vertices in relations $(2, 1, 1)$ and $(2, 2, 1)$, respectively, to $x, y, z$.

# In[6]:


p.tripleEquations(3, 1, 1, params={"alpha": (2, 1, 1), "beta": (2, 2, 1)})


# From $[1\ 1\ 3] \ge 0$ and $[3\ 3\ 2] \ge 0$, it follows that $60 - 5\alpha/2 \le \beta \le 8$. Using this and $[3\ 3\ 1] \ge 0$, we obtain $20.8 \le a \le 22$, but since $\alpha$ and $\beta$ must clearly be even integers, it follows that there are two solutions with $\alpha = 22$ and $\beta \in \{6, 8\}$. In both cases we have $[3\ 3\ 1] = 0$, implying that $w$ cannot be in relation $1$ with $z$ for any choice of $w, x, y, z$ as above.
# 
# We now compute the triple intersection numbers with respect to $w, x, y$. The parameters $\gamma$ and $\delta$ will denote the numbers of vertices in relations $(1, 1, 2)$ and $(1, 2, 2)$, respectively, to $w, x, y$.

# In[7]:


p.tripleEquations(3, 3, 3, params={"gamma": (1, 1, 2), "delta": (1, 2, 2)})


# From $[1\ 3\ 3] \ge 0$ and $[2\ 3\ 3] \ge 0$, it follows that $\gamma, \delta \le 44/3$, while $[3\ 3\ 3] \ge 0$ implies $\gamma + \delta \ge 28$. Since $\gamma$ and $\delta$ must clearly be even integers, it follows that there is only one solution with $\gamma = \delta = 14$ and therefore $[1\ 1\ 1] = 3$. This would imply the existence of a vertex $z$ as above that is in relation $1$ with $w$ - a contradiction! We thus conclude that a $Q$-polynomial association scheme with Krein array $\{24, 20, 36/11; 1, 30/11, 24\}$ **does not exist**.