Quadratic Residue Codes over ${\mathbb{Z}}_{121}$

In this paper, we construct a special family of cyclic codes, known as quadratic residue codes of prime length $p\equiv \pm 1(\mathrm{m}\mathrm{o}\mathrm{d}44),$ $p\equiv \pm 5(\mathrm{m}\mathrm{o}\mathrm{d}44),$ $p\equiv \pm 7(\mathrm{m}\mathrm{o}\mathrm{d}44),$ $p\equiv \pm 9(\mathrm{m}\mathrm{o}\mathrm{d}44)$ and $p\equiv \pm 19(\mathrm{m}\mathrm{o}\mathrm{d}44)$ 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 $[77,7,44]\mathrm{.}$