Mysql – How to design a table that each rows have 5K boolean attributes

I have about 2M rows and each row looks like the following.

244 true false ... true

-> One integer column(V) and about 5K boolean columns(B1, B2, …, B5K) associated to the integer.

Due to the limitation of the maximum number of columns that I can have for a row, I have separated the boolean columns(attributes) in a separate table.

Table V:

idx_V value_V
--------------
1     244  

...

Table B:

idx_V idx_B value_B
--------------------
1     1     true
1     2     false
...
1     5K    true
...

This design works alright when I try to find V's that match one boolean column. For example, finding V's where the 2nd boolean attribute is true:

select value_V 
where VT.idx_A = BT.idx_A 
    and idx_B = 2 
    and value_B = true 
from V_Table as VT 
    and B_Table as BT

But the query becomes awful when I have to find V's that match a multiple boolean columns, sometimes even for all 5K columns, like finding V's with B1=true, B2=false, B3=true, … and B5K=false.

My primary use of the tables would be the following 2:

1. Find V's that x1th, x2th and xnth boolean columns are false/true (n can be anything between 1 and 5K)
2. Sublists:
   • Find the sequence of the boolean columns for a specific V: T F T T F F ...
   • Find other V's that match the sequence found in 2-A

I'm thinking about constructing a varchar[5K] field to store the boolean sequence to do 2 but it seems like there's just too much waste in space since each boolean only requires just 1 bit but I'm allocating a byte.

What would be the best way to go about this?