For Coleman, the case is all about finding more votes to surpass Democrat Al Franken's 312-vote lead.
In their 62-page brief, Coleman's attorneys say the three-judge panel didn't open and count enough ballots because it applied a standard that was too strict.
Midway through the trial, the panel ruled some categories of ballots couldn't be added to the tally because they didn't fulfill all the requirements of Minnesota law. But some similar ballots had already been counted by the state canvassing board and added to the vote total.
Coleman's brief says the panel violated the constitution and asks the state supreme court to send the case back to the panel and order it to look at more rejected ballots through a more generous lens.
Coleman attorney Ben Ginsberg said the panel should have counted ballots as long as voters substantially complied with the law. That could mean opening and counting at least another 1,400 ballots.
"That standard was wrong because it was not applied in any jurisdiction in counting their votes on Election Day, and it also runs counter to Minnesota's rulings in the past of its tradition of enfranchising voters rather than disenfranchising voters," he said.
Coleman's brief says if the Minnesota Supreme Court doesn't order the panel to count more votes, the justices should declare that it's impossible to determine a winner. That would presumably prompt a revote. In addition, Coleman argues the panel erred when it declined to order inspections at precincts where Coleman alleged that poll workers double counted ballots; and that 132 ballots from a Franken-leaning precinct should be subtracted from the tally since they disappeared after Election Day.
Democrat Al Franken's lawyers said they would respond to Coleman's arguments in their own brief, which they must file by Monday, May 11.
On Tuesday, Sen. Arlen Specter of Pennsylvania announced he was abandoning the GOP for the Democratic Party. Minnesota's empty senate seat went from being just one seat in the Senate to the pivotal vote that could block Republican opposition to President Obama's initiatives.
With Specter's move across the aisle, Democrats have 59 votes in the senate. They need Al Franken to reach a filibuster-proof, supermajority of 60.
The state Supreme Court will hear oral arguments on the case June 1.