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

# # Nonexistence of a $Q$-polynomial association scheme with Krein array $\{2r^2-1, 2r^2-2, r^2+1; 1, 2, r^2-1\}$ and $r$ odd
# 
# We will show that a $Q$-polynomial association scheme with Krein array $\{2r^2-1, 2r^2-2, r^2+1; 1, 2, r^2-1\}$ and $r$ odd does not exist. This Krein array is feasible for all $r \ge 2$. For $r = 2^j$, this Krein array is realized by the duals of Kasami codes with minimum distance $5$.

# In[1]:


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


# Such a scheme would have $4r^4$ vertices.

# In[2]:


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


# Since the scheme is $Q$-polynomial, we have $q^1_{13} = q^1_{31} = q^3_{11} = 0$. Additionally, we have $q^1_{11} = 0$.

# In[3]:


[p.q[1, 1, 1], p.q[1, 1, 3], p.q[1, 3, 1], p.q[3, 1, 1]]


# The intersection numbers can be checked to be nonnegative and integral for both even and odd values of $r \ge 2$.

# In[4]:


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


# Let us check that the case $r = 3$ satisfies the known feasibility conditions. We skip the family nonexistence check since the Krein array of the members of family with $r$ odd is already included.

# In[5]:


p.subs(r == 3).check_feasible(skip=["family"])


# We now compute the triple intersection numbers with respect to three vertices $x, y, z$ such that $x$ is in relation $1$ with $y$ and $z$, and $y$ is in relation $2$ with $z$. Note that we have $p^2_{11} = r(r+2)(r^2-1)/4 > 0$ for all $r \ge 2$, so such triples must exist. The parameters $\alpha, \beta, \gamma, \delta$ will denote the number of vertices in relations $(1, 3, 1), (2, 3, 3), (3, 1, 1), (3, 3, 3)$, respectively, to $x, y, z$.

# In[6]:


S112 = p.tripleEquations(1, 1, 2, params={'alpha': (1, 3, 1), 'beta': (2, 3, 3), 'gamma': (3, 1, 1), 'delta': (3, 3, 3)})
S112[1, 2, 3]


# We see that the above triple intersection number can only be integral when $r$ is even. We thus conclude that a $Q$-polynomial association scheme with Krein array $\{2r^2-1, 2r^2-2, r^2+1; 1, 2, r^2-1\}$ and $r$ odd **does not exist**.