def solve():
    n, k = map(int, input().split())
    if n == 2:
        if k == 2:
            print("Yes")
            print("##")
            print("..")
        else:
            print("No")
        return
    
    if k <= 2 or 2*k > n*n:
        print("No")
        return
    
    print("Yes")
    if k <= n:
        print("#"*(k-1) + "."*(n-k+1))
        print("#" + "."*(n-1))
        for i in range(n - 2):
            print("."*n)
        return
    
    c = [["."] * n for i in range(n)]
    p, q = divmod(2*k, n)
    lim = [p+1] * q + [p] * (n-q) 
    for i in range(n):
        if i & 1:
            for j in range(lim[i] - 2):
                c[i][j] = "#"
        c[i][lim[i]-2] = "#"
        if i != n-1:
            c[i][lim[i+1]-2] = "#"
    
    s = sum(c[i].count("#") for i in range(n))
    for i in range(k - s):
        c[0][i] = "#"
    
    for i in range(n):
        print("".join(c[i]))
    return

t = int(input())
for i in range(t):
    solve()