While in Phd I developed a lot of code that I want to turn into start-up. Can I do this?

I'm not aware of any blanket prohibition, anywhere, that say that you can't use code developed during a PhD for a commercial venture - but most Universities in the US will have some kind of intellectual property agreement in place to state what the process is for doing this.

You will need to check with officials at your University to see what agreements you are subject to - but you should go a step farther than this. Specifically, you should see what departments might exist that can actually help you.

In the US at least - and I'm sure many other institutions around the world - there are departments specializing in intellectual property, "knowledge transfer", corporate spin-offs, start-ups, entrepreneurship, etc. These can be a rich source of support and networking, if they are available to you, even if you have intellectual property agreements that assign various rights to the University.

Universities want to earn money for these projects, as I'm sure you do as well, and 100% of $0 is $0. Universities often heavily facilitate the process of converting this work into money, and it is not abnormal for even a place that has heavily restricted intellectual rights assignments agreements to assign use rights and royalty agreements that have zero cost for the first $200,000+ US dollars in income. There are often even startup grants available, accelerators (even if they are not called that - places that provide space and equipment to help starting businesses), introductions to investors, showcases, and more. You won't know until you ask and look for them, as where they are located tends to vary heavily by institution.

For the exact nature of the agreements that apply to you, we can't say; I've seen everything from "the student owns their own work" to "anyone funded assigns all rights and ownership to the University", and I'm sure there are agreements everywhere in between. You are right to check them out in advance of a major decision, but this is so specific that you'll need to talk with multiple people are your specific institution to see what applies to you. You may also need to speak with an outside legal professional (lawyer experienced in this area, etc) to verify, but your local administrators are your best first point of contact to see how things work.

You should ask officially, as others (notably BrianH) answered. At some places and countries, your university -or research organization funding your PhD work- will even help you (perhaps with a paid leave and/or some business incubator). At other places, it will forbid that and you could be sued. Perhaps your advisor might be helpful (so consider speaking with him about that startup idea, if you are on good terms).

In France, where I live, getting help on a startup with a PhD idea is probably easy (thru clusters like Systematic, BPI France, etc...). The official stance is that France is wanting more of that (and provide dedicated funds to help that), and it is likely to be the same in most European countries.

However, I recommend finishing your PhD (with 3 years of work on it you probably are not very far from that), get the official doctoral degree, and later on work on your startup (with approval, in some official setting which might ask you to give some parts of your company to the University or some other licensing terms, and perhaps help, from your University). An abandoned PhD work is a failure.

Even if your startup works, having the PhD (really) shows that you are able to end a difficult challenge. And if your startup fails (which is likely, since most of them do) the PhD is a useful blanket (or safety net) to have. 5 to 10 years later from now, having a PhD on your business card could make a lot of difference (and perhaps earlier: I guess that banks will more easily lend you money if your startup has the blessing of your University or research institution, whatever that means).

Conversely, dropping your PhD work right now, so close to finishing it, is probably a very stupid thing (and could send the wrong message to future VCs or potential clients : that you cannot be trusted because you are impulsive). Think a lot before committing that.

BTW having a working code on some research problem is really different from being able to sell it successfully (and requires different skills). And a research prototype software is quite different from a commercial product.

I've been through this a few times, but note that I'm not a lawyer. This is just advice to get you started on what you can look further into.

First of all code per se is covered by copyright not patents, and the copyright is probably owned by your employer (in a typical employment agreement). If the govt funds it you may be required to make it available to other researchers. If you never sell or publish the code you're using at your startup, no one generally knows or cares you're using it. Of course I've heard of people trying to sell their company get burned when the lawyers doing due diligence submits the code for copyright check.

The invention described by the code would presumably be patentable (because everything is these days), but whatever you have already published is no longer patentable in many countries. And in the US it will no longer be able to be patented one year after publication. This is all similarly owned by your employer if still patentable. After you publish it of course you and anyone else can do it.

One option is to see if you can get the University to patent it and have your company license it from them. This would actually be a great option as it adds prestige to your company. Plus the university's lawyers handle the legal stuff and protect/enforce the patent. This will also help you get funding. VC's will want to see you have protection for your IP (and conversely they will of course be turned off by potential IP issues).

Another option is to "engineer around" it by changing everything somewhat. Presumably you as the expert on the topic can imagine different ways of doing the same thing.

A final option is to not worry about the university and see if you can get anywhere on your own first. Then if you meet with success, you can now afford to deal with the IP issues. This is more the bootstrap option if you'll be scrounging one customer at a time and no one even knows what you're up to. Frankly the odds are you will find you need to change what you are doing anyway. Every customer tends to need a totally custom product oftentimes.