NEW YORK—The Manhattan District Attorney's office on Monday appealed a judge's decision that absolved Sergey Aleynikov, a programmer accused of stealing sophisticated trading software from Goldman Sachs Group Inc. and exonerated by two separate courts.

The appeal means Mr. Aleynikov's long-running legal woes aren't yet over, and raises the prospect he still could be convicted of stealing the investment bank's high-frequency trading code.

Earlier in July, New York state Justice Daniel P. Conviser dismissed a conviction of Mr. Aleynikov, saying prosecutors hadn't shown enough evidence to support a jury's May verdict that Mr. Aleynikov made unlawful use of secret scientific material. Prosecutors filed court documents Monday asking the New York court's appellate division to overturn Justice Conviser's order, which would effectively reinstate the jury's verdict.

Justice Conviser ruled a decades-old law written before the advent of the computer age didn't apply to Mr. Aleynikov's actions. The order was a rebuke to prosecutors in the office of Manhattan District Attorney Cyrus R. Vance Jr., who pushed the case in state court after a similar case was thrown out at the federal level three years ago.

Mr. Aleynikov's lawyer, Kevin Marino, assailed the appeal in a written statement on Monday.

"Mr. Vance's decision to prosecute Mr. Aleynikov for copying Goldman Sachs's computer source code after the Second Circuit ruled that he'd spent a year in federal prison for conduct that was not a federal crime was reprehensible," Mr. Marino said. "To compound that decision by appealing Justice Conviser's ruling that Mr. Aleynikov's source code copying was also not a state crime…is inexcusable."

Mr. Aleynikov was convicted on charges of stealing the bank's "secret sauce" trading code in federal court in 2010, but was acquitted by an appellate court in 2012 after spending a year in prison. Less than a year after his federal case was thrown out, Mr. Vance's office charged him under state laws.

It is rare for state prosecutors to bring a criminal case related to the same conduct after a federal conviction is overturned—and some lawyers in the defense bar at the time described the move as overly aggressive. Mr. Vance's office has said it interpreted the appellate court's ruling that federal laws didn't apply to Mr. Aleynikov's conduct as an invitation for state charges.

Mr. Aleynikov's second trial in state court was marked by esoteric legal disputes, with jurors appearing to struggle with how to interpret and apply laws designed well before the computer age, asking several times for charges to be reread and for definitions of various words.

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