#line 2 "/Users/noya2/Desktop/Noya2_library/template/template.hpp"
using namespace std;

#include<bits/stdc++.h>
#line 1 "/Users/noya2/Desktop/Noya2_library/template/inout_old.hpp"
namespace noya2 {

template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p){
    os << p.first << " " << p.second;
    return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p){
    is >> p.first >> p.second;
    return is;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v){
    int s = (int)v.size();
    for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
    return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v){
    for (auto &x : v) is >> x;
    return is;
}

void in() {}
template <typename T, class... U>
void in(T &t, U &...u){
    cin >> t;
    in(u...);
}

void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u){
    cout << t;
    if (sizeof...(u)) cout << sep;
    out(u...);
}

template<typename T>
void out(const vector<vector<T>> &vv){
    int s = (int)vv.size();
    for (int i = 0; i < s; i++) out(vv[i]);
}

struct IoSetup {
    IoSetup(){
        cin.tie(nullptr);
        ios::sync_with_stdio(false);
        cout << fixed << setprecision(15);
        cerr << fixed << setprecision(7);
    }
} iosetup_noya2;

} // namespace noya2
#line 1 "/Users/noya2/Desktop/Noya2_library/template/const.hpp"
namespace noya2{

const int iinf = 1'000'000'007;
const long long linf = 2'000'000'000'000'000'000LL;
const long long mod998 =  998244353;
const long long mod107 = 1000000007;
const long double pi = 3.14159265358979323;
const vector<int> dx = {0,1,0,-1,1,1,-1,-1};
const vector<int> dy = {1,0,-1,0,1,-1,-1,1};
const string ALP = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
const string alp = "abcdefghijklmnopqrstuvwxyz";
const string NUM = "0123456789";

void yes(){ cout << "Yes\n"; }
void no(){ cout << "No\n"; }
void YES(){ cout << "YES\n"; }
void NO(){ cout << "NO\n"; }
void yn(bool t){ t ? yes() : no(); }
void YN(bool t){ t ? YES() : NO(); }

} // namespace noya2
#line 2 "/Users/noya2/Desktop/Noya2_library/template/utils.hpp"

#line 6 "/Users/noya2/Desktop/Noya2_library/template/utils.hpp"

namespace noya2{

unsigned long long inner_binary_gcd(unsigned long long a, unsigned long long b){
    if (a == 0 || b == 0) return a + b;
    int n = __builtin_ctzll(a); a >>= n;
    int m = __builtin_ctzll(b); b >>= m;
    while (a != b) {
        int mm = __builtin_ctzll(a - b);
        bool f = a > b;
        unsigned long long c = f ? a : b;
        b = f ? b : a;
        a = (c - b) >> mm;
    }
    return a << std::min(n, m);
}

template<typename T> T gcd_fast(T a, T b){ return static_cast<T>(inner_binary_gcd(std::abs(a),std::abs(b))); }

long long sqrt_fast(long long n) {
    if (n <= 0) return 0;
    long long x = sqrt(n);
    while ((x + 1) * (x + 1) <= n) x++;
    while (x * x > n) x--;
    return x;
}

template<typename T> T floor_div(const T n, const T d) {
    assert(d != 0);
    return n / d - static_cast<T>((n ^ d) < 0 && n % d != 0);
}

template<typename T> T ceil_div(const T n, const T d) {
    assert(d != 0);
    return n / d + static_cast<T>((n ^ d) >= 0 && n % d != 0);
}

template<typename T> void uniq(std::vector<T> &v){
    std::sort(v.begin(),v.end());
    v.erase(unique(v.begin(),v.end()),v.end());
}

template <typename T, typename U> inline bool chmin(T &x, U y) { return (y < x) ? (x = y, true) : false; }

template <typename T, typename U> inline bool chmax(T &x, U y) { return (x < y) ? (x = y, true) : false; }

template<typename T> inline bool range(T l, T x, T r){ return l <= x && x < r; }

} // namespace noya2
#line 8 "/Users/noya2/Desktop/Noya2_library/template/template.hpp"

#define rep(i,n) for (int i = 0; i < (int)(n); i++)
#define repp(i,m,n) for (int i = (m); i < (int)(n); i++)
#define reb(i,n) for (int i = (int)(n-1); i >= 0; i--)
#define all(v) (v).begin(),(v).end()

using ll = long long;
using ld = long double;
using uint = unsigned int;
using ull = unsigned long long;
using pii = pair<int,int>;
using pll = pair<ll,ll>;
using pil = pair<int,ll>;
using pli = pair<ll,int>;

namespace noya2{

/*　~ (. _________ . /)　*/

}

using namespace noya2;


#line 2 "c.cpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/graph/grid.hpp"

#line 5 "/Users/noya2/Desktop/Noya2_library/graph/grid.hpp"
#include <ranges>

namespace noya2 {

struct grid {
    int h, w;
    grid (int _h = 0, int _w = 0) : h(_h), w(_w) {}
    int idx(int x, int y) const {
        return x * w + y;
    }
    std::pair<int,int> pos(int id) const {
        return {id / w, id % w};
    }
    bool operator()(int x, int y) const {
        return 0 <= x && x < h && 0 <= y && y < w;
    }
    bool operator()(int id) const {
        return operator()(id / w, id % w);
    }
    static constexpr std::array<std::pair<int,int>, 4> dxys = {std::pair<int,int>{0, 1}, {-1, 0}, {0, -1}, {1, 0}};
    auto adj4(int x, int y) const {
        return dxys | std::views::transform([x, y](std::pair<int,int> dxy){
            return std::pair<int,int>(dxy.first + x, dxy.second + y);
        }) | std::views::filter([this](std::pair<int,int> xy){
            return operator()(xy.first, xy.second);
        });
    }
    auto adj4(int id) const {
        return adj4(id / w, id % w) | std::ranges::views::transform([this](std::pair<int,int> xy){
            return idx(xy.first, xy.second);
        });
    }
};

} // namespace noya2
#line 4 "c.cpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/data_structure/dsu.hpp"

#line 6 "/Users/noya2/Desktop/Noya2_library/data_structure/dsu.hpp"

namespace noya2{

struct dsu {
  public:
    dsu() : _n(0), _cc(0) {}
    dsu(int n) : _n(n), _cc(n), parent_or_size(n, -1) {}

    int merge(int a, int b) {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        int x = leader(a), y = leader(b);
        if (x == y) return x;
        if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);
        parent_or_size[x] += parent_or_size[y];
        parent_or_size[y] = x;
        _cc--;
        return x;
    }

    bool same(int a, int b) {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        return leader(a) == leader(b);
    }

    int leader(int a) {
        assert(0 <= a && a < _n);
        if (parent_or_size[a] < 0) return a;
        return parent_or_size[a] = leader(parent_or_size[a]);
    }

    int size(int a) {
        assert(0 <= a && a < _n);
        return -parent_or_size[leader(a)];
    }

    std::vector<std::vector<int>> groups() {
        std::vector<int> leader_buf(_n), group_size(_n);
        for (int i = 0; i < _n; i++) {
            leader_buf[i] = leader(i);
            group_size[leader_buf[i]]++;
        }
        std::vector<std::vector<int>> result(_n);
        for (int i = 0; i < _n; i++) {
            result[i].reserve(group_size[i]);
        }
        for (int i = 0; i < _n; i++) {
            result[leader_buf[i]].push_back(i);
        }
        result.erase(
            std::remove_if(result.begin(), result.end(),
                           [&](const std::vector<int>& v) { return v.empty(); }),
            result.end());
        return result;
    }

    int group_count() const {
        return _cc;
    }

  private:
    int _n, _cc;
    // root node: -1 * component size
    // otherwise: parent
    std::vector<int> parent_or_size;
};

} // namespace noya2
#line 6 "c.cpp"

void solve(){
    int n, k; in(n,k);
    if (k == 1){
        no();
        return ;
    }
    if (k > n*n/2){
        no();
        return ;
    }
    if (n == 2){
        assert(k == 2);
        yes();
        out("##");
        out("..");
        return ;
    }
    if (k == 2){
        no();
        return ;
    }
    vector<string> ans(n,string(n,'.'));
    auto say = [&](){
        yes();
        rep(i,n){
            out(ans[i]);
        }
        grid gr(n,n);
        dsu uf(n*n);
        rep(i,n) rep(j,n){
            if (ans[i][j] == '.') continue;
            assert(ans[i][j] == '#');
            for (auto [ni, nj] : gr.adj4(i,j)){
                if (ans[ni][nj] != '#') continue;
                uf.merge(gr.idx(i,j),gr.idx(ni,nj));
            }
        }
        int leader = -1;
        rep(i,n) rep(j,n){
            if (ans[i][j] != '#') continue;
            int l = uf.leader(gr.idx(i,j));
            if (leader == -1){
                leader = l;
            }
            assert(leader == l);
        }
        rep(i,n) rep(j,n){
            if (ans[i][j] == '#') continue;
            for (auto [ni, nj] : gr.adj4(i,j)){
                if (ans[ni][nj] == '#'){
                    ans[i][j] = '!';
                }
            }
        }
        int c4 = 0, d4 = 0;
        rep(i,n) rep(j,n){
            if (ans[i][j] == '#'){
                c4++;
            }
            if (ans[i][j] == '!'){
                d4++;
            }
        }
        // out(c4,d4,k);
        assert(c4 == k && d4 == k);
    };
    if (k <= n){
        rep(i,k-1){
            ans[0][i] = '#';
        }
        ans[1][0] = '#';
        say();
        return ;
    }
    if (k <= n*2){
        rep(i,n-(k-n)){
            ans[0][i] = '#';
        }
        int rem = 2*(k-n);
        reb(i,n){
            if (rem == 0) break;
            ans[1][i] = '#';
            rem--;
        }
        reb(i,n){
            if (rem == 0) break;
            ans[2][i] = '#';
            rem--;
        }
        say();
        return ;
    }
    for (int v = k*2-1; v >= k*2-n; v--){
        int i = v / n, j = v % n;
        ans[i-1][j] = '#';
    }
    if (k*2 % n != 0){
        int v = k*2-1 - n;
        int i = v / n, j = v % n;
        ans[i-1][j] = '#';
    }
    int ilim = (k*2-n)/n-1;
    for (int i = 1; i <= ilim; i++){
        ans[i][n-1] = '#';
    }
    for (int i = 1; i <= ilim; i += 3){
        rep(j,n){
            ans[i][j] = '#';
        }
    }
    int cnt = 0;
    rep(i,n) rep(j,n){
        if (ans[i][j] == '#'){
            cnt++;
        }
    }
    rep(i,n) rep(j,n){
        if (ans[i][j] == '#') continue;
        if (cnt == k) break;
        ans[i][j] = '#';
        cnt++;
    }
    say();
}

int main(){
    int t = 1; in(t);
    while (t--) { solve(); }
}