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

arXiv:2603.24689v1 Announce Type: new
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 $[77,7,44].$