Neither the A340-600 or the 777-300ER with normal pax loads is operating at max range, i.e. the tanks aren't full, so payload can be traded for greater or lesser range. Performance gains over predictions on the 773ER give more range for a given payload, more payload for a given range, or more of each. According to Boeing:
The additional thrust [of the GE90-115B] increases the 777-300ER’s maximum take-off weight to 759,600 pounds – almost 100,000 pounds more than the 777-300 – with virtually no difference in handling characteristics during takeoff, flight and landing.
That power comes at very little expense. Fuel mileage testing shows a 1 percent improvement in fuel efficiency over original predictions. Such an improvement can reduce by 106,400 gallons (402,724 liters) annually the amount of fuel one airplane uses. That’s enough to power 130 automobiles for one year.
It also increases the airplane’s range by 75 nautical miles (139 kilometers) to 7,495 nautical miles (13,881 kilometers); or it increases payload by 2,400 pounds (1,089 kilograms) -- passenger or cargo – on a 7,000 nautical mile (12,964 kilometer) flight.
Airbus' advertised range for the A346 is 7,500 nm, giving it a statistically insignificant range advantage of 5 nm.
If you're referring to the HGW versions, the 773ER's tanks are not full at typical payloads. With a higher MTOW, more fuel can be carried for a given payload, resulting in longer range. The higher takeoff weights on both aircraft will shift out the payload-range curve, allowing either greater range, more payload, or both.
[Edited 2003-10-29 04:06:12]
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