In making a brick oven, the dome for igloo and vault for barrel oven is a tricky part to build. The bricks are held in place by cement, but before the oven dries up and the cement becomes firm enough, an additional reinforcing material is required. A formwork of wood is the most common used however, this formwork is expensive. It is therefore recommended that you use locally sourced formworks for example, for the barrel oven, you can use a drum cut to take the shape of the barrel oven.
Further information: Build a pizza oven foundation of out wood, Build a wood burning clay pizza oven, and Make your own barrel brick pizza oven and outdoor setting