c++ - abstract class and alternative to virtual constructor

Let say I have the following code:

class Block{
private:
  data Data;
public:
  data getData();
};

Actually, there are several ways to build a block, but always with the same member Data and method getData(), the only difference is how to build the block. In other words, the only difference is the constructor...

Instead of writing a different class for each building process, I could factorize parts of my code, defining and declaring getData in an abstract class, if there were such thing as a virtual constructor in c++ that I could write differently for each derived class corresponding to a different building process.

I do not have a lot experience for this kind of things, so I wondered if there was an alternative to a virtual constructor ? or may be a different way to do this factorization ?

PS: I am aware of http://ift.tt/2s7oBWN but it seems quite complex regarding what I want to do, which seems quite common... I just want to factorize shared code between several classes, which corresponds to everything except the constructor. And I want to force new classes corresponding to other building processes to implement a new constructor.

More details about my particular situation:

I have an algorithm where I use blocks and it does not depend on their building process, so I have implemented the algorithm using a template argument to represent a block indifferently of its building process. But I use a few methods and its constructor, so I need my classes representing blocks to all have the same kind of methods I need and the same constructor to use them as a template argument of my algorithm implementation. That is why I thought of abstract class, to force a newly implemented class representing blocks to have the methods and the constructor I need in the algorithm I implemented. May be it is a bad design pattern and that is why I am stuck...