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

Abstract

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 $[77,7,44].$