def main(n, k):
    if 2 * k > n * n:
        return []
    Ans = [[0 for _ in range(n)] for _ in range(n)]
    if k < n:
        if k < 3:
            return []
        elif k == 3:
            Ans[0][0] = 1
            Ans[0][1] = 1
            Ans[1][0] = 1
            return Ans
        elif k < n:
            h = k // 2
            w = k - h
            for y in range(h):
                Ans[y][0] = 1
                k -= 1
            for x in range(1, w):
                Ans[h-1][x] = 1
                k -= 1
            y, x = h-2, 1
            while k:
                Ans[y][x] = 1
                k -= 1
                x += 1
                if x == w:
                    x = 1
                    y -= 1
            return Ans
    if k == 2 * n:
        for y in range(n):
            Ans[y][1] = 1
            Ans[y][2] = 1
        return Ans
    elif k == 2 * n + 1:
        Ans[0][1] = 1
        Ans[0][3] = 1
        Ans[1][3] = 1
        for y in range(1, n):
            Ans[y][1] = 1
            Ans[y][2] = 1
        return Ans
    K = 2*k
    t = K // n
    s = K % n
    x = t-1
    cnt = 0
    while x >= 0:
        for y in range(s):
            cnt += 1
            Ans[y][x] = 1
        for y in range(s, n):
            if 0 <= x-1:
                cnt += 1
                Ans[y][x-1] = 1
            else:
                cnt += 1
                Ans[y][x] = 1
        x -= 3
    if s == 0:
        r = t-1
    else:
        r = t
    y = (s-1) % n
    for x in range(r):
        if not Ans[y][x]:
            cnt += 1
            Ans[y][x] = 1
    ama = k - cnt
    y = x = 0
    while ama:
        if not Ans[y][x]:
            Ans[y][x] = 1
            ama -= 1
        y += 1
        if y == n:
            y = 0
            x += 1
    cnt = 0
    for y in range(n):
        for x in range(n):
            if Ans[y][x]:
                cnt += 1
    assert cnt == k
    return Ans
    
for _ in range(int(input())):
    n, k = map(int, input().split())
    Ans = main(n, k)
    if Ans:
        print("Yes")
        Ans = [["#" if Ans[y][x] else "." for x in range(n)] for y in range(n)]
        for ans in Ans:
            print("".join(ans))
    else:
        print("No")