add unique code required to register

Add a field on your registration form

<label for="secret_reg_code">Secret Code: </label>
<input type="text" name="secret_reg_code" id="secret_reg_code" value="" />

then in your plugin file or in function.php add

define('MY_SECRET_REG_CODE', 'fdsfgadsfgdf');

function check_for_reg_secret_code($errors, $sanitized_user_login, $user_email) {
  if (
    defined('MY_SECRET_REG_CODE') &&
    trim($_POST['secret_reg_code']) != MY_SECRET_REG_CODE
  ) {
    $errors->add(
      'secret_code_error',
      __('<strong>ERROR</strong>: Enter a valid code postal code.', 'your_text_domain')
    );
  }
  return $errors;
}
add_filter('registration_errors', 'check_for_reg_secret_code', 10, 3);

A more ‘flexible’ way is use WP Settings Api to create an option ‘secret_reg_code’ where to store the secret code and then the function became:

function check_for_reg_secret_code($errors, $sanitized_user_login, $user_email) {
  $code = get_option('secret_reg_code');
  if ( $code && trim($_POST['secret_reg_code']) != $code) {
    ...

In this way is easy change code periodically.

For more details see Codex for registration_errors filter hook

Edit

To add the field on the standard registration form is possible to use the register_form hook:

function add_secret_code_registration_field() {
    ?>
    <p>
    <label for="secret_reg_code">Secret Code<br />
    <input type="text" name="secret_reg_code" id="secret_reg_code" class="input" value="" size="25" /></label>
    </p>
    <?php
}
add_action('register_form','add_secret_code_registration_field');