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.

中文翻译：

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一系列排列不变的量子码

我们构建了一个新的排列不变码族，可以纠正任何 $t\ge 1$ 的 $t$ 泡利错误。我们还表明，新家族中的代码可以纠正量子缺失错误以及自发衰变错误。我们的构造包含一些先前已知的排列不变量子代码作为特殊情况，它们也允许横向门。在许多情况下，新家族中的代码比先前已知的泡利错误和删除的显式排列不变代码更短。此外，我们的新代码系列包括新的 $((4,2,2))$ 最佳单删除校正代码。作为一个单独的结果，我们将排列不变码的条件概括为将 $t$ 泡利错误从 $t=1$ 的先前已知结果纠正为任意数量的错误。对于较小的$t$，这些条件可用于通过计算机构造新的代码示例。