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

# # Nonexistence of a $Q$-polynomial association scheme with Krein array $\{(r^2+1)/2, (r^2-1)/2, (r^2+1)^2/(2r(r+1)), (r-1)(r^2+1)/(4r), (r^2+1)/(2r); 1, (r-1)(r^2+1)/(2r(r+1)), (r+1)(r^2 + 1)/(4r), (r-1)(r^2+1)/(2r), (r^2+1)/2\}$ and $r \equiv 3 \pmod{4}$
# 
# We show that a $Q$-polynomial association scheme with Krein array
# \begin{multline*}
# \{(r^2+1)/2, (r^2-1)/2, (r^2+1)^2/(2r(r+1)), (r-1)(r^2+1)/(4r), (r^2+1)/(2r); \\
# 1, (r-1)(r^2+1)/(2r(r+1)), (r+1)(r^2 + 1)/(4r), (r-1)(r^2+1)/(2r), (r^2+1)/2\}
# \end{multline*}
# and $r \equiv 3 \pmod{4}$ does not exist. For $r \equiv 1 \pmod{4}$ a prime power, this Krein array is realized by a scheme derived by [Moorhouse and Williford](https://arxiv.org/pdf/1504.01067.pdf) from a double cover of the $C_2(r)$ dual polar graph.

# In[1]:


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


# Such a scheme would have $2(r+1)(r^2+1)$ vertices.

# In[2]:


r = var("r")
p = drg.QPolyParameters([(r^2+1)/2, (r^2-1)/2, (r^2+1)^2/(2*r*(r+1)), (r-1)*(r^2+1)/(4*r), (r^2+1)/(2*r)],
                        [1, (r-1)*(r^2+1)/(2*r*(r+1)), (r+1)*(r^2 + 1)/(4*r), (r-1)*(r^2+1)/(2*r), (r^2+1)/2])
p.order(factor=True, simplify=True)


# This scheme is $Q$-bipartite, so it has $q^h_{ij} = 0$ whenever $h+i+j$ is odd, or $h, i, j$ do not satisfy the triangle inequality. Additionally, we have $q^4_{55} = q^5_{45} = q^5_{54} = 0$.

# In[3]:


p.kreinParameters(factor=True, simplify=2)


# The intersection numbers can be checked to be nonnegative and integral for all odd values of $r \ge 3$.

# In[4]:


p.pTable(factor=True, simplify=True)


# By the absolute bound for $(1, 1)$, such a scheme is only feasible when $r \ge 5$.

# In[5]:


p.check_absoluteBound(expand=True, factor=True)


# Let us check that the smallest admissible case with $r \equiv 3 \pmod{4}$, namely $r = 7$, satisfies the known feasibility conditions. We skip the family nonexistence check since the Krein array of the members of family with $r \equiv 3 \pmod{4}$ is already included.

# In[6]:


p.subs(r == 7).check_feasible(skip=["family"])


# We now compute the triple intersection numbers with respect to three vertices $x, y, z$ mutually in relation $1$. Note that we have $p^1_{11} = (r-1)/2 > 0$ for all $r \ge 5$, so such triples must exist.

# In[7]:


p.dualEigenmatrix(expand=True, factor=True, simplify=True)
p.tripleEquations(1, 1, 1)


# We see that there is a single solution, which can only be integral if $r \equiv 1 \pmod{4}$. We thus conclude that a $Q$-polynomial association scheme with Krein array
# \begin{multline*}
# \{(r^2+1)/2, (r^2-1)/2, (r^2+1)^2/(2r(r+1)), (r-1)(r^2+1)/(4r), (r^2+1)/(2r); \\
# 1, (r-1)(r^2+1)/(2r(r+1)), (r+1)(r^2 + 1)/(4r), (r-1)(r^2+1)/(2r), (r^2+1)/2\}
# \end{multline*}
# and $r \equiv 3 \pmod{4}$ **does not exist**.