Find binary numbers that generate all n-bit binary numbers using at most one XOR; see https://math.stackexchange.com/questions/4611907.

Question4611907.java

import java.util.Random;

public class Question4611907 {
    final static int k = 52;
    final static int n = 10;
    final static int m = 1 << n;
    
    final static Random random = new Random();

    static int count;
    static int [] counts = new int [m];
    static int [] x = new int [k];
    
    static void increment (int x) {
        if (counts [x]++ == 0)
            count++;
    }
    
    static void decrement (int x) {
        if (--counts [x] == 0)
            count--;
    }
    
    static void clear (int i) {
        decrement (x [i]);
        for (int j = 0;j < k;j++)
            if (j != i)
                decrement (x [i] ^ x [j]);
    }
    
    static void set (int i) {
        increment (x [i]);
        for (int j = 0;j < k;j++)
            if (j != i)
                increment (x [i] ^ x [j]);
    }
    
    static void randomize (int i) {
        x [i] = random.nextInt(m);
    }
    
    public static void main (String [] args) {
        increment (0);
        
        for (int i = 0;i < k;i++) {
            randomize (i);
            increment (x [i]);
            for (int j = 0;j < i;j++)
                increment (x [i] ^ x [j]);
        }

        
        int l = 0;
        int max = 0;
        
        for (double beta = 0;count < m;beta += .00000001) {
            for (int i = 0;i < k;i++) {
                int oldCount = count;
                int oldX = x [i];
                clear (i);
                randomize (i);
                set (i);
                if (count < oldCount && Math.random() > Math.exp(beta * (count - oldCount))) {
                    clear (i);
                    x [i] = oldX;
                    set (i);
                }
                max = Math.max(max,count);
            }

            if ((++l & 0x7fff) == 0) {
                System.out.println(beta + " : " + max + " / " + count);
            }
        }
        
        for (int i = 0;i < k;i++) {
            System.out.write((i & 7) == 0 ? '\n' : ' ');
            System.out.print(String.format("%" + n + "s", Integer.toBinaryString(x [i])).replace(' ', '0'));
        }
    }
}