def solve():
    N, K = [int(s) for s in input().split()]

    if N == 2:
        if K == 2:
            print("Yes")
            print("##")
            print("..")
        else:
            print("No")
        return

    if K <= 2 or N**2 // 2 < K:
        print("No")
        return

    if K <= N:
        print("Yes")
        print("#" * (K - 1) + "." * (N - K + 1))
        print("#" + "." * (N - 1))
        for _ in range(N - 2):
            print("." * N)
        return

    grid = [["."] * N for _ in range(N)]

    div = (2 * K) // N
    mod = (2 * K) % N

    for i in range(N):
        if mod > 0 and i < mod:
            grid[div - 1][i] = "#"
        else:
            grid[div - 2][i] = "#"

    for i in range(div - 1):
        for j in range(1, N - 1, 3):
            grid[i][j] = "#"
        grid[i][N - 2] = "#"

    if mod > 0:
        grid[mod - 1][div - 2] = "#"

    remain = K - sum(sum(int(grid[i][j] == "#") for j in range(N)) for i in range(N))
    for i in range(N):
        for j in range(N):
            if remain > 0 and grid[i][j] == ".":
                grid[i][j] = "#"
                remain -= 1

    print("Yes")
    for row in grid:
        print(*row, sep="")


if __name__ == "__main__":
    T = int(input())

    for _ in range(T):
        solve()