We construct a new family of permutationally invariant codes that correct $t$ Pauli errors for any $t\ge 1$. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new $((4,2,2))$ optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct $t$ Pauli errors from the previously known results for $t=1$ to any number of errors. For small $t$, these conditions can be used to construct new examples of codes by computer.

翻译：我们构造了一类新的置换不变码家族，能够纠正任意$t\ge 1$个泡利错误。同时证明该家族中的码还能纠正量子删除错误和自发衰变错误。我们的构造将部分已知的置换不变量子码作为特例包含在内，这些码也支持横向门操作。在许多情况下，新家族中的码比此前已知的最佳显式置换不变码（针对泡利错误和删除错误）具有更短的码长。此外，新码家族包含一个最优的$((4,2,2))$单删除纠正码。作为独立结果，我们将置换不变码纠正$t$个泡利错误的条件从已知的$t=1$情形推广到任意错误数。对于较小的$t$，这些条件可用于通过计算机构造新的码实例。