Say a company gives a week long assignment during the hiring process of a Software Engineer role and ask for the project´s codebase to remain private.
Can they morally and legally ask to keep that private? Who has right / ownership of this project?
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If you agreed to keep it private, you are bound to do so.
This answer pulls information from the following sources regarding the enforceability of non-disclosure agreements.
Non-Disclosure Agreements and Confidentiality Agreements are common elements of the hiring process. It would not be unusual if they required you to sign one in consideration for an interview.
If you signed one, you should consider it a binding contract and absolutely enforceable.
Regarding project ownership. If the employer had an existing codebase and asked you to contribute to it in exchange for being considered for a job opening, they would own the project, the code and the work product you contributed.
All this assumes an actual job exists and they are not tricking you just to get free labor. If they are, then that would be fraud and contracts are not enforceable if their is "fraud in the inducement."
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It depends on what the explicit agreement was but generally speaking based on the facts you've given yes they can ask for it to remain private. In consideration for being a job candidate you agreed to do work and keep the work private. Their half of the contract was to evaluate you as a job candidate and your half was to do the work and keep it private.
Did the other party fulfill their part?