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The Laws of Cryptography with Java Code.

Here is Verhoeff's scheme using the dihedral group using special permutations. Notice that all adjacent transpositions are detected.

// ErrorDetection.java: base class for single-digit error detection
public class ErrorDetection {

   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("? ");
      for (int i = 1; i < a.length; i++) {
      	 System.out.print(a[i]);
      	 if (i%5 == 0) System.out.print(" ");
      }
      System.out.println();
   }
}

// VerhoeffErrorDetection.java: Implement Verhoeff's decimal error detection
public class VerhoeffErrorDetection extends ErrorDetection {

   private static int[][] op= {
      {0, 1, 2, 3, 4, 5, 6, 7, 8, 9},
      {1, 2, 3, 4, 0, 6, 7, 8, 9, 5},
      {2, 3, 4, 0, 1, 7, 8, 9, 5, 6},
      {3, 4, 0, 1, 2, 8, 9, 5, 6, 7},
      {4, 0, 1, 2, 3, 9, 5, 6, 7, 8},
      {5, 9, 8, 7, 6, 0, 4, 3, 2 ,1},
      {6, 5, 9, 8, 7, 1, 0, 4, 3, 2},
      {7, 6, 5, 9, 8, 2, 1, 0, 4, 3},
      {8, 7, 6, 5, 9, 3, 2, 1, 0, 4},
      {9, 8, 7, 6, 5, 4, 3, 2, 1, 0} };

   private static int[] inv = {0, 4, 3, 2, 1, 5, 6, 7, 8, 9};
   private static int[][] F = new int[8][];
   private static int[] F0 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9};
   private static int[] F1 = {1, 5, 7, 6, 2, 8, 3, 0, 9, 4};
   
   public VerhoeffErrorDetection() {
      F[0] = F0;
      F[1] = F1;
      for (int i = 2; i < 8; i++) {
         F[i] = new int[10];
         for (int j = 0; j < 10; j++)
            F[i][j] = F[i-1] [F[1][j]];
      }
   }
   
   public static int insertCheck(int[] a) {
      int check = 0;
      for (int i = 1; i < a.length; i++)
         check = op[check][ F[i % 8][a[i]] ];
      a[0] = inv[check];
      return a[0];
   }
   
   public static boolean doCheck(int[] a) {
      int check = 0;
      for (int i = 0; i < a.length; i++)
         check = op[check][ F[i % 8][a[i]] ];
      if (check != 0) return false;
      else return true;
   }

   // main function
   public static void main (String[] args) {
      VerhoeffErrorDetection v = new VerhoeffErrorDetection();
      int[] a = new int[15];
      boolean checkFlag = false;
      for (int i = 1; i < a.length; i++)
         a[i] = (int)(Math.random() * 10.0);
      VerhoeffErrorDetection.printUnchecked(a);
      VerhoeffErrorDetection.insertCheck(a);
      VerhoeffErrorDetection.printArray(a);
      System.out.println(VerhoeffErrorDetection.doCheck(a));
      a[4] = (a[4] + 1)%10;

      VerhoeffErrorDetection.printArray(a);
      System.out.println(VerhoeffErrorDetection.doCheck(a));

      // test all adjacent transpositions
      System.out.println("\nTest all adjacent transpositions");
      for (int p1 = 0; p1 < 10; p1++)
         for (int p2 = 0; p2 < 10; p2++) {
            if (p1 != p2) {
               a[8] = p1; a[9] = p2;
               VerhoeffErrorDetection.insertCheck(a);
               // interchange
               a[8] ^= a[9];
               a[9] ^= a[8];
               a[8] ^= a[9];
               if (VerhoeffErrorDetection.doCheck(a)) {
                  System.out.println("Warning: Interchange of " + p1 +
                     " and " + p2 + " not detected");
                  checkFlag = true;
               }
            }
         }
      if (checkFlag)
         System.out.println("At least one transposition undetected");
      else
         System.out.println("All transpositions detected");

   } // end of main
} 

? 75787 12372 9429
1 75787 12372 9429
true
1 75797 12372 9429
false

Test all adjacent transpositions
All transpositions detected