The gate teleportation protocol (https://arxiv.org/abs/quant-ph/9908010) has a step where it utilizes cat state measurement of transversal operators. Since cat states can not detect phase errors it is suggested that one should repeat cat state measurement $r$ times for a large enough $r$ and then take the majority vote. Is there any reference for how large $r$ should be, say for a code of distance $d$?
I think you're not getting a lot of answers because qec has advanced since 1999 and no one would do fault tolerant gate teleportation by repeatedly preparing vulnerable cat states. You're far better off preparing states that are directly protected, instead of bolting majority voting on top. Majority voting is horribly inefficient.
For example, suppose you are making your cat states using the surface code and you are picking the code distance to use for protection against phase error. You can either use 5 votes (tolerate 2 failures), 3 votes (tolerate 1 failure), or 1 vote (tolerate no failures). Suppose you're willing to accept a logical error rate of one in a trillion, which is achieved by a patch width of 36. We're considering making the patches skinnier, lowering their width, increasing their failure rate, and making up for the difference using majority voting.
For the 1 vote case, you use a code distance of 36 one time. Call that a hardware time cost of 36*1 = 36.
For the 3 vote case, you can use half the width (square rooting the per-vote error rate to 1 in a million), so you pay a cost of 36/2 three times. Call that a hardware time cost of 18*3 = 54.
For the 3 vote case, you can use a third the width (cube rooting the per-vote error rate to 1 in 10 thousand), so you pay a cost of 36/3 five times. Call that a hardware time cost of 12*5 = 60.
You see the problem? The more majority voting we use, the more hardware time we are spending on the task! It's more efficient to just use the error correction that was already present; in this case the surface code. Concatenating isn't always bad, sometimes you can stack two codes and get benefits, but majority voting is almost always a bad thing to concatenate. Doing K votes has to compete with using K times as many qubits to define your logical qubits, which is a huge opportunity cost.