LOD means Line-out dock, a dock that sends line-out signal from an source to the amp, and line-out is ALWAYS an analog signal. The problem with Lightning connector is that it only has digital signal, so it doesn't have any line-out or analog signal to send, and therefore you can't make a LOD out of it. The best you can do is making a 'digital cable' for some DAC that support iDevices. The genuine Apple Lightning-to-30pins adapter is NOT a LOD, but actually a miniature DAC. Therefore using it can give you line-out signal, though it might not be better than double amping the headphone-out.
So if you can get a Lightning digital cable, can it support any DAC? Unfortunately the answer is no. To really get iDevice to detect external DAC, you need to have the MFi chip in the DAC, which you will need to get a license from Apple to use. that's why many external DAC that has MFi chip costs over US$500 (there are a few cheaper one, in the US$300 region). But if you have a iOS7 device, you can use the camera connection kit (CCK) to enable the USB host function hiding inside iOS7, which will let you use most USB DAC that doesn't have the MFi chip, as long as the USB DAC itself doesn't draw too much power from the iOS7 device (and if you ever wonder - yes, the Lightning-to-30pins adapter had the MFi chip built-in as well).
