#include <cassert> #include <cmath> #include <cstdint> #include <cstdio> #include <cstdlib> #include <cstring> #include <algorithm> #include <bitset> #include <complex> #include <deque> #include <functional> #include <iostream> #include <limits> #include <map> #include <numeric> #include <queue> #include <random> #include <set> #include <sstream> #include <string> #include <unordered_map> #include <unordered_set> #include <utility> #include <vector> using namespace std; using Int = long long; template <class T1, class T2> ostream &operator<<(ostream &os, const pair<T1, T2> &a) { return os << "(" << a.first << ", " << a.second << ")"; }; template <class T> ostream &operator<<(ostream &os, const vector<T> &as) { const int sz = as.size(); os << "["; for (int i = 0; i < sz; ++i) { if (i >= 256) { os << ", ..."; break; } if (i > 0) { os << ", "; } os << as[i]; } return os << "]"; } template <class T> void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; } template <class T> bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; } template <class T> bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; } #define COLOR(s) ("\x1b[" s "m") //////////////////////////////////////////////////////////////////////////////// template <unsigned M_> struct ModInt { static constexpr unsigned M = M_; unsigned x; constexpr ModInt() : x(0U) {} constexpr ModInt(unsigned x_) : x(x_ % M) {} constexpr ModInt(unsigned long long x_) : x(x_ % M) {} constexpr ModInt(int x_) : x(((x_ %= static_cast<int>(M)) < 0) ? (x_ + static_cast<int>(M)) : x_) {} constexpr ModInt(long long x_) : x(((x_ %= static_cast<long long>(M)) < 0) ? (x_ + static_cast<long long>(M)) : x_) {} ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; } ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; } ModInt &operator*=(const ModInt &a) { x = (static_cast<unsigned long long>(x) * a.x) % M; return *this; } ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); } ModInt pow(long long e) const { if (e < 0) return inv().pow(-e); ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b; } ModInt inv() const { unsigned a = M, b = x; int y = 0, z = 1; for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast<int>(q) * z; y = z; z = w; } assert(a == 1U); return ModInt(y); } ModInt operator+() const { return *this; } ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; } ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); } ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); } ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); } ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); } template <class T> friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); } template <class T> friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); } template <class T> friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); } template <class T> friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); } explicit operator bool() const { return x; } bool operator==(const ModInt &a) const { return (x == a.x); } bool operator!=(const ModInt &a) const { return (x != a.x); } friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; } }; //////////////////////////////////////////////////////////////////////////////// constexpr unsigned MO = 998244353; using Mint = ModInt<MO>; #ifndef LIBRA_ALGEBRA_NIMBER_H_ #define LIBRA_ALGEBRA_NIMBER_H_ #include <assert.h> //////////////////////////////////////////////////////////////////////////////// namespace nim { using u16 = unsigned short; using u32 = unsigned; using u64 = unsigned long long; // G16: primitive root in F_(2^16) // G16\^3 = 2^15 constexpr u16 G16 = 10279U; u16 expBuffer[4 * (1 << 16) + 4]; u16 *exp = expBuffer + (2 * (1 << 16) + 4), *exp3 = exp + 3, *exp6 = exp + 6; int log[1 << 16]; u64 tabSq[4][1 << 16], tabSqrt[4][1 << 16], tabSolveQuad1[4][1 << 16]; // L: power of 2 // (a0 + 2^l a1) \* (b0 + 2^l b1) // = (a0\*b0 \+ 2^(l-1)\*a1\*b1) + 2^l (a0\*b1 \+ a1\*b0 \+ a1\*b1) template <int L> inline u64 mulSlow(u64 a, u64 b) { static constexpr int l = L >> 1; const u64 a0 = a & ((1ULL << l) - 1), a1 = a >> l; const u64 b0 = b & ((1ULL << l) - 1), b1 = b >> l; const u64 a0b0 = mulSlow<l>(a0, b0); return (a0b0 ^ mulSlow<l>(1ULL << (l - 1), mulSlow<l>(a1, b1))) | (a0b0 ^ mulSlow<l>(a0 ^ a1, b0 ^ b1)) << l; } template <> inline u64 mulSlow<1>(u64 a, u64 b) { return a & b; } // 2^31 \* a inline u32 mul31(u32 a) { const u16 a0 = a, a1 = a >> 16; const u16 a01 = a0 ^ a1; return exp6[log[a1]] | (u32)exp3[log[a01]] << 16; } inline u16 mul(u16 a, u16 b) { return exp[log[a] + log[b]]; } inline u32 mul(u32 a, u32 b) { const u16 a0 = a, a1 = a >> 16; const u16 b0 = b, b1 = b >> 16; const u16 a01 = a0 ^ a1; const u16 b01 = b0 ^ b1; const u16 a0b0 = mul(a0, b0); return (a0b0 ^ exp3[log[a1] + log[b1]]) | (u32)(a0b0 ^ mul(a01, b01)) << 16; } inline u64 mul(u64 a, u64 b) { const u32 a0 = a, a1 = a >> 32; const u32 b0 = b, b1 = b >> 32; const u32 a01 = a0 ^ a1; const u32 b01 = b0 ^ b1; const u32 a0b0 = mul(a0, b0); return (a0b0 ^ mul31(mul(a1, b1))) | (u64)(a0b0 ^ mul(a01, b01)) << 32; } inline u16 sq(u16 a) { return tabSq[0][a]; } inline u32 sq(u32 a) { const u16 a0 = a, a1 = a >> 16; return tabSq[0][a0] ^ tabSq[1][a1]; } inline u64 sq(u64 a) { const u16 a0 = a, a1 = a >> 16, a2 = a >> 32, a3 = a >> 48; return tabSq[0][a0] ^ tabSq[1][a1] ^ tabSq[2][a2] ^ tabSq[3][a3]; } inline u16 sqrt(u16 a) { return tabSqrt[0][a]; } inline u32 sqrt(u32 a) { const u16 a0 = a, a1 = a >> 16; return tabSqrt[0][a0] ^ tabSqrt[1][a1]; } inline u64 sqrt(u64 a) { const u16 a0 = a, a1 = a >> 16, a2 = a >> 32, a3 = a >> 48; return tabSqrt[0][a0] ^ tabSqrt[1][a1] ^ tabSqrt[2][a2] ^ tabSqrt[3][a3]; } // (a0 + 2^l a1) \* (b0 + 2^l b1) = 1 // <=> [ a0 2^(l-1)\*a1 ] \* [ b0 ] = [ 1 ] // [ a1 a0\+a1 ] [ b1 ] [ 0 ] inline u16 inv(u16 a) { assert(a); return exp[((1 << 16) - 1) - log[a]]; } inline u32 inv(u32 a) { assert(a); const u16 a0 = a, a1 = a >> 16; const u16 a01 = a0 ^ a1; const u16 d = inv((u16)(mul(a0, a01) ^ exp3[log[a1] + log[a1]])); return mul(d, a01) | (u32)mul(d, a1) << 16; } inline u64 inv(u64 a) { assert(a); const u32 a0 = a, a1 = a >> 32; const u32 a01 = a0 ^ a1; const u32 d = inv(mul(a0, a01) ^ mul31(sq(a1))); return mul(d, a01) | (u64)mul(d, a1) << 32; } // f(x) := x\^2 \+ x // bsr(x\^2) = bsr(x) // f: {even in [0, 2^L)} -> [0, 2^(L-1)): linear isom. // f(x0 + 2^l x1) = (f(x0) \+ 2^(l-1)\*x1\^2) + 2^l f(x1) template <int L> inline u64 solveQuad1Slow(u64 a) { static constexpr int l = L >> 1; assert(!(a >> (L - 1))); const u64 a0 = a & ((1ULL << l) - 1), a1 = a >> l; const u64 x1 = solveQuad1Slow<l>(a1); const u64 b0 = a0 ^ mul(1ULL << (l - 1), sq(x1)); const u64 s = b0 >> (l - 1); return solveQuad1Slow<l>(b0 ^ s << (l - 1)) | (x1 ^ s) << l; } template <> inline u64 solveQuad1Slow<1>(u64 a) { assert(!a); return 0; } // x\^2 \+ x \+ a = 0 // solutions: x, x \+ 1 inline u64 solveQuad1(u64 a) { assert(!(a >> 63)); const u16 a0 = a, a1 = a >> 16, a2 = a >> 32, a3 = a >> 48; return tabSolveQuad1[0][a0] ^ tabSolveQuad1[1][a1] ^ tabSolveQuad1[2][a2] ^ tabSolveQuad1[3][a3]; } // x\^2 \+ a\*x \+ b = 0 // solutions: x, x \+ a inline bool isSolvableQuad(u64 a, u64 b) { return !(mul(inv(sq(a)), b) >> 63); } inline u64 solveQuad(u64 a, u64 b) { return a ? mul(a, solveQuad1(mul(inv(sq(a)), b))) : sqrt(b); } struct Preparator { Preparator() { exp[0] = 1; for (int i = 1; i < (1 << 16) - 1; ++i) exp[i] = mulSlow<16>(exp[i - 1], G16); for (int i = (1 << 16) - 1; i < 2 * (1 << 16); ++i) exp[i] = exp[i - ((1 << 16) - 1)]; for (int i = 0; i < (1 << 16) - 1; ++i) log[exp[i]] = i; log[0] = -(1 << 16) - 2; for (int e = 0; e < 64; ++e) { const u64 x = mul(1ULL << e, 1ULL << e); for (int i = 0; i < 1 << (e & 15); ++i) tabSq[e >> 4][i | 1 << (e & 15)] = tabSq[e >> 4][i] ^ x; } for (int e = 0; e < 64; ++e) { u64 x = 1ULL << e; for (int j = 0; j < 63; ++j) x = sq(x); for (int i = 0; i < 1 << (e & 15); ++i) tabSqrt[e >> 4][i | 1 << (e & 15)] = tabSqrt[e >> 4][i] ^ x; } for (int e = 0; e < 63; ++e) { const u64 x = solveQuad1Slow<64>(1ULL << e); for (int i = 0; i < 1 << (e & 15); ++i) tabSolveQuad1[e >> 4][i | 1 << (e & 15)] = tabSolveQuad1[e >> 4][i] ^ x; } } } preparator; } // namespace nim //////////////////////////////////////////////////////////////////////////////// #endif // LIBRA_ALGEBRA_NIMBER_H_ struct Nim64 { unsigned long long x; constexpr Nim64() : x(0ULL) {} constexpr Nim64(unsigned x_) : x(x_) {} constexpr Nim64(unsigned long long x_) : x(x_) {} constexpr Nim64(int x_) : x(x_) {} constexpr Nim64(long long x_) : x(x_) {} Nim64 &operator+=(const Nim64 &a) { x ^= a.x; return *this; } Nim64 &operator-=(const Nim64 &a) { x ^= a.x; return *this; } Nim64 &operator*=(const Nim64 &a) { x = nim::mul(x, a.x); return *this; } // TODO: operator/=, pow, inv Nim64 operator+() const { return *this; } Nim64 operator-() const { return *this; } Nim64 operator+(const Nim64 &a) const { return (Nim64(*this) += a); } Nim64 operator-(const Nim64 &a) const { return (Nim64(*this) -= a); } Nim64 operator*(const Nim64 &a) const { return (Nim64(*this) *= a); } // TODO: operator/ explicit operator bool() const { return x; } bool operator==(const Nim64 &a) const { return (x == a.x); } bool operator!=(const Nim64 &a) const { return (x != a.x); } friend std::ostream &operator<<(std::ostream &os, const Nim64 &a) { return os << a.x; } }; int M, N; Nim64 A[20][20]; int main() { /* using namespace nim; cerr << (mul(3ULL - 1, 12ULL - 1) ^ mul(6ULL - 1, 2ULL - 1) ^ mul(9ULL - 1, 2ULL - 1)) << endl; cerr << (mul(5ULL - 1, 12ULL - 1) ^ mul(10ULL - 1, 2ULL - 1) ^ mul(15ULL - 1, 2ULL - 1)) << endl; */ for (; ~scanf("%d%d", &N, &M); ) { for (int i = 0; i < M; ++i) for (int j = 0; j < N; ++j) { scanf("%llu", &A[i][j].x); --A[i][j].x; } vector<int> rs(1 << N); vector<vector<vector<Nim64>>> as(1 << N); rs[0] = 0; as[0] = vector<vector<Nim64>>(M, vector<Nim64>(N)); for (int i = 0; i < M; ++i) for (int j = 0; j < N; ++j) as[0][i][j] = A[i][j]; for (int h = 0; h < N; ++h) { for (int p = 0; p < 1 << h; ++p) { // cerr<<"h = "<<h<<", p = "<<p<<", rs[p] = "<<rs[p]<<", as[p] = "<<as[p]<<endl; int &r = rs[p | 1 << h] = rs[p]; auto &a = as[p | 1 << h] = as[p]; for (int i = r; i < M; ++i) if (a[i][0]) { swap(a[r], a[i]); break; } if (r < M && a[r][0]) { const Nim64 s = nim::inv(a[r][0].x); for (int j = 1; j < (int)a[r].size(); ++j) a[r][j] *= s; for (int i = r + 1; i < M; ++i) for (int j = 1; j < (int)a[i].size(); ++j) a[i][j] -= a[i][0] * a[r][j]; ++r; } for (int i = 0; i < M; ++i) { as[p][i].erase(as[p][i].begin()); a[i].erase(a[i].begin()); } } } // cerr<<"rs = "<<rs<<endl; Mint ans = 0; for (int p = 0; p < 1 << N; ++p) { const int n = __builtin_popcount(p); ans += ((N-n)&1?-1:+1) * Mint(2).pow(64).pow(n - rs[p]); } printf("%u\n", ans.x); } return 0; }