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The Laws of Cryptography with Java Code.

Here is a Java program that simulates random double errors. There are two input parameters on the command line: the number of simulation trials to run, and the number of digits to use. The program identifies various outcomes:

Thus, with 18 digits, only 14.4% of double errors get "corrected" as if there were a single error, while with 121 digits 81.5% are miscorrected in this way.

// H11ED.java: Implement the mod 11 Hamming code 
public class H11ED {

   public static int[] inv = {0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1};

   public static int totalTrials, misCorrected = 0,
      errorZero = 0, toTen = 0, subscriptRange = 0, allZero = 0;
   public static int arraySize;

   // Using the sum1 check sum, if have error e and check1 result c,
   //    then pos[e][c] gives the position in error (modulo 11),
   //    using the first check equation.
   //   If the error is e and check11 result is c,
   //    then pos[e][c] gives the value position/11.
   public static int[][] pos = {
         {0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0},  
         {0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10},
         {0,  6,  1,  7,  2,  8,  3,  9,  4, 10,  5},
         {0,  4,  8,  1,  5,  9,  2,  6, 10,  3,  7},
         {0,  3,  6,  9,  1,  4,  7, 10,  2,  5,  8},
         {0,  9,  7,  5,  3,  1, 10,  8,  6,  4,  2},
         {0,  2,  4,  6,  8, 10,  1,  3,  5,  7,  9},
         {0,  8,  5,  2, 10,  7,  4,  1,  9,  6,  3},
         {0,  7,  3, 10,  6,  2,  9,  5,  1,  8,  4},
         {0,  5, 10,  4,  9,  3,  8,  2,  7,  1,  6},
         {0, 10,  9,  8,  7,  6,  5,  4,  3,  2,  1}};

   public static void printArray(int[] a) {
      for (int i = 0; i < a.length; i++) {
         if (a[i] == 10)
            System.out.print("X ");
         else
            System.out.print(a[i]);
         if (i%5 == 0) System.out.print(" ");
      }
      System.out.println();
   }
   
   public static void printUnchecked(int[] a) {
      System.out.print("? "); // position 0
      System.out.print("?"); // position 1
      for (int i = 2; i < a.length; i++) {
         if (i == 11) System.out.print("?"); // position 11
         else System.out.print(a[i]);
         if (i%5 == 0) System.out.print(" ");
      }
      System.out.println();
   }

   public static void insertCheck(int[] a) {
      a[1] = inv[sum1NoCheck(a)];
      a[11] = inv[sum11NoCheck(a)];
      a[0] = inv[sum0NoCheck(a)];
      // if (!doCheck(a))
      //   System.out.println("Failure in insertCheck");
   }
   
   public static boolean doCheck(int[] a) {
      int error = sum0(a); // amount of error
      int check1 = sum1(a);
      int check11 = sum11(a);
      if (error == 0 && check1 == 0 && check11 == 0) {
         allZero++;
         return true;
      }
      if (error == 0) {
          // System.out.println("Double error: check 0 is zero");
          errorZero++;
          return false; // a double error
      }
      int position = pos[error][check11]*11 + pos[error][check1];
      if (position >= a.length) {
         subscriptRange++;
         // System.out.println("Doouble error:" +
         //    " subscript error, position: " + position +
         //    ", error: " + error + ", check1: " + check1 +
         //    ", check11: " + check11);
         // System.exit(0);
         return false;
      }
      a[position] = (a[position] - error + 11)%11;
      if (a[position] == 10 && (position != 0 && position != 1 &&
            position != 11)) {
         // System.out.println("Double error: corrected to 10");
         toTen++;
         return false;
      }
      // System.out.println("Position " + position + " corrected to " + 
      //    a[position]);
      misCorrected++;
      return true;
   }

   public static int sum0(int[] a) {
      int check = 0;
      for (int i = 0; i < a.length; i++)
         check = (check + a[i])%11;
      return check;
   }

   public static int sum0NoCheck(int[] a) {
      int check = 0;
      for (int i = 1; i < a.length; i++)
         check = (check + a[i])%11;
      return check;
   }

   public static int sum1(int[] a) {
      int check = 0;
      for (int i = 0; i < a.length; i++)
         check = (check + (i%11)*a[i])%11;
      return check;
   }

   public static int sum1NoCheck(int[] a) {
      int check = 0;
      for (int i = 2; i < a.length; i++)
         check = (check + (i%11)*a[i])%11;
      return check;
   }
   
   public static int sum11(int[] a) {
      int check = 0;
      for (int i = 0; i < a.length; i++)
         check = (check + ((i/11)%11)*a[i])%11;
      return check;
   }
   
   public static int sum11NoCheck(int[] a) {
      int check = 0;
      for (int i = 12; i < a.length; i++)
         check = (check + ((i/11)%11)*a[i])%11;
      return check;
   }
   
   // main function
   public static void main (String[] args) {
      totalTrials = Integer.parseInt(args[0]); // total num of trials
      arraySize   = Integer.parseInt(args[1]); // size of array
      int[] a = new int[arraySize];
      int loc1, loc2;
      boolean checkFlag = false;
      // start with random word
      for (int i = 2; i < a.length; i++)
         if (i != 11) a[i] = (int)(Math.random() * 10.0);
      H11ED.insertCheck(a);
      for (int i = 0; i < totalTrials; i++) {
         // try random pair of errors, choose 2 distinct random ints
         loc1 = (int)(Math.random() * a.length);
         do {
                loc2 = (int)(Math.random() * a.length);
         } while (loc1 == loc2);
         // System.out.print("Locs: " + loc1 + " " + loc2);
         // System.out.print(", Values: " + a[loc1] + " " + a[loc2]);
         // make a random pair of changes
         a[loc1] = (a[loc1] + (int)(Math.random() * 9.0 + 1.0))%10;
         a[loc2] = (a[loc2] + (int)(Math.random() * 9.0 + 1.0))%10;
         // System.out.println(", New Values: " + a[loc1] + " " + a[loc2]);
         H11ED.doCheck(a);
         // System.out.println(doCheck(a));
      }
      if (totalTrials != (misCorrected + errorZero + toTen +
            subscriptRange + allZero))
         System.out.println("Count Off");
      System.out.println("Total: " + totalTrials +
         ", errorZero: "    + errorZero +
         ", toTen: "        + toTen +
         ", subscript: "    + subscriptRange +
         ", misCorrected: " + misCorrected +
         ", allZero: "      + allZero);
   } // end of main
} 

% myjava H11ED 10000000 18
Total: 10000000, errorZero: 919248, toTen: 130462, subscript: 7501457, misCorrected: 1444074, allZero: 4759
% myjava H11ED 1000000 121
Total: 1000000, errorZero: 100002, toTen: 85080, subscript: 0, misCorrected: 814827, allZero: 91