Why would that be? I've looked at the rules, and as far as I comprehend them I should be able to fly the AA codeshare. Here's the specifics:

SYD-CHC: Flight is QF45, codeshare is AA7345, I see L7 availability on 18th May
AKL-SYD: Flight is QF164, codeshare is AA 7324, L7 availability on 15th July
SYD-HNL: Flight is QF3, codeshare is AA 7359, L7 availability on 17th July.

Is all that correct, or am I not able to fly on the codeshare at all?