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We use the Haar function system in order to study the $L_2$ discrepancy of aclass of digital $(0,n,2)$-nets. Our approach yields exact formulas for thisquantity, which measures the irregularities of distribution of a set of pointsin the unit interval. We will obtain such formulas not only for the classicaldigital nets, but also for shifted and symmetrized versions thereof. The basicidea of our proofs is to calculate all Haar coefficents of the discrepancyfunction exactly and insert them into Parseval's identity. We will also discussreasons why certain (symmetrized) digital nets fail to achieve the optimalorder of $L_2$ discrepancy and use the Littlewood-Paley inequality in order toobtain results on the $L_p$ discrepancy for all $p\in (1,\infty)$.

Finding exact formulas for the $L_2$ discrepancy of digital $(0,n,2)$-nets via Haar functions