You're so close. Instead of a blank default value in your ternary statement, just put your default value (and ditch your invalid repetition of the value
attribute).:
<input type="text"
value="<?php echo (isset($_POST["postcode"]) ? $_POST["postcode"] : '119900'); ?>"
id="postcode" name="postcode" size=6 maxlength=6/>
You can also shorten this a bit:
<input type="text" value="<?= ($_POST["postcode"]) ?: '119900'); ?>"
id="postcode" name="postcode" size=6 maxlength=6/>
(For anyone complaining about the short tag usage, the minimum supported version of PHP is 5.4 which has them enabled by default. So unless you are targeting older versions of PHP and/or plan to distribute the code this is fine to do).