In this paper, we construct a special family of cyclic codes, known as quadratic residue codes of prime length \( p \equiv \pm 1 \pmod{44} ,\) \( p \equiv \pm 5 \pmod{44} ,\) \( p \equiv \pm 7 \pmod{44} ,\) \( p \equiv \pm 9 \pmod{44} \) and \( p \equiv \pm 19 \pmod{44} \) over $\mathbb{Z}_{121}$ by defining them using their generating idempotents. Furthermore, the properties of these codes and extended quadratic residue codes over $\mathbb{Z}_{121}$ are discussed, followed by their Gray images. Also, we show that the extended quadratic residue code over $\mathbb{Z}_{121}$ possesses a large permutation automorphism group generated by shifts, multipliers, and inversion, making permutation decoding feasible. As examples, we construct new codes with parameters $[55,5,33]$ and $[77,7,44].$

翻译：本文通过定义生成幂等元，在环$\mathbb{Z}_{121}$上构造了一类特殊的循环码族，即素数长度 \( p \equiv \pm 1 \pmod{44} \)，\( p \equiv \pm 5 \pmod{44} \)，\( p \equiv \pm 7 \pmod{44} \)，\( p \equiv \pm 9 \pmod{44} \) 和 \( p \equiv \pm 19 \pmod{44} \) 的二次剩余码。进一步地，探讨了这些码及$\mathbb{Z}_{121}$上扩展二次剩余码的性质，并给出它们的Gray像。此外，我们证明$\mathbb{Z}_{121}$上的扩展二次剩余码拥有由移位、乘法和逆运算生成的大置换自同构群，这使得置换译码成为可行。作为示例，我们构造了参数为$[55,5,33]$和$[77,7,44]$的新码。