By filing 28 lawsuits in 13 states seeking nearly $1 million in overdue child support, the U.S. Justice Department opened a promising and more aggressive new phase this week in the war on deadbeat parents.

But even more vigorous crackdowns and more imaginative reforms will be needed if this campaign is to be as successful as it can and should be.The new prosecutions are being pursued under a newly funded law that makes it a federal offense for a divorced parent to willfully fail to pay more than $5,000 in court-ordered support for a child living in another state.

First-time offenders can be slapped with a $5,000 fine and put behind bars for up to six months. Repeat offenders can get up to two years in jail and a $250,000 fine.

In addition to the 28 lawsuits filed this week, another 200 cases are in the pipeline. Even though the sums involved in the current crackdown are large, they represent only a portion of child support owed nationally. The Justice Department estimates deadbeat parents owe a total of $34 billion in support for more than 14 million children.

These figures may tempt some deadbeats to think the odds favor their continuing to get away with short-changing their children. If so, they should think again. Already, the first stages of what could eventually become a nationwide computer network is enabling officials to track virtually anyone with a Social Security number, locate the absent parent and garnish their wages. In California alone, this system has enabled the state to collect $11.2 million from in-state parents in just five months. Soon California expects to collect as much as $103 million from 11,600 parents living out of state.

Other states also should seriously consider a promising reform adopted by Maine, which has started to lift the drivers' licenses of parents who refuse to pay overdue child support. This threat has yielded $11.5 million in back payments in only a year.

Another remedy would help, too. One big reason so many parents are delinquent in paying child support is that their after-tax income is so low. The main reason for this sad situation, in turn, is that increases in the personal tax exemption for children have lagged far behind the rise in taxes and inflation over the past four decades. A sharp hike in this tax break is clearly in order.

By all means, federal and state governments should get tougher on deadbeat parents. But let's apply the carrot as well as the stick.