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