#include <bits/stdc++.h>
using namespace std;

void yes(const vector<vector<char>> grid) {
    int length = grid.size();
    cout << "Yes" << "\n";
    for (int i = 0; i < length; i++) {
        for (int j = 0; j < length; j++) {
            cout << grid[i][j];
        }
        cout << "\n";
    }
}

void no() {
    cout << "No" << "\n";
}

void solve() {
    int N, K;
    cin >> N >> K;

    if (N == 2) {
        if (K == 2) {
            vector<vector<char>> grid{
                {'#', '#'},
                {'.', '.'}
            };
            yes(grid);
        } else {
            no();
        }
        return;
    }

    if (K <= 2 || (N * N) / 2 < K) {
        no();
        return;
    }

    vector grid(N, vector<char>(N, '.'));

    if (K <= N) {
        for (int j = 0; j < K - 1; j++) {
            grid[0][j] = '#';
        }
        grid[1][0] = '#';
        yes(grid);
        return;
    }

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

    for (int j = 0; j < N; j++) {
        if (mod > 0 && j < mod) {
            grid[div - 1][j] = '#';
        } else {
            grid[div - 2][j] = '#';
        }
    }

    for (int i = 0; i < div - 1; i++) {
        for (int j = 1; j < N - 1; j += 3) {
            grid[i][j] = '#';
        }
        grid[i][N - 2] = '#';
    }

    if (mod > 0) {
        grid[div - 2][mod - 1] = '#';
    }

    int remain = K;
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            remain -= grid[i][j] == '#' ? 1 : 0;
        }
    }

    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            if (remain > 0 && grid[i][j] == '.') {
                grid[i][j] = '#';
                remain -= 1;
            }
        }
    }
    yes(grid);
}

int main() {
    cin.tie(nullptr);
    ios::sync_with_stdio(false);

    int T;
    cin >> T;

    while (T--) solve();
}