Look at the subforum for diy here on Audioholics or avsforum.com, or subs on diyaudio.com, etc for sub designs/plans. They'll be specifying not only the box details but also the specific drivers the box is based on. The options for driver/box you'll need to sort for yourself somewhat, but think the research would give you some ideas to go on within your budget. A good quality pro amp with 2 channels is easier than using plate amps (plus I just don't want my amp in/on the box myself)....something from Crown or QSC (or if you want to spend some money maybe SpeakerPower).
I've used Stereo Integrity drivers (18s) as well as a coupla Infinity drivers (12s) but that's been a while since I researched those and took note of what's out there both for designs and availability of drivers (Stereo Integrity has scaled back ops since then due owner's health issues, but depends what size driver you want somewhat; the Infinity drivers I used are no longer available afaik), plus I built them several years back so no particular plans that I've run across recently as haven't been looking.
I went sealed myself for a coupla reasons, one being I have limited experience in woodworking and finishing (and am easy to please on the finishing, I just stained some baltic birch ply for some, used duratex on others), size, weight, and how many subs I was willing to use to make up for the lack of efficiency a ported would bring. I'm using Crown XLS amps and minidsp units (2x4HD would be what I'd get now) but with a ported sub you may not need much in the way of eq if no protective high pass filter below tuning is needed and your Audyssey alone may even do the trick.
Might be a bit of a rabbit hole compared to just buying something, tho