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

TL;DR

This paper constructs quadratic residue codes over Z_{121} for specific prime lengths, analyzes their properties, and demonstrates their automorphism groups, leading to new codes with notable parameters.

Contribution

It introduces a new family of quadratic residue codes over Z_{121} for primes satisfying certain congruences, and explores their properties and automorphisms.

Findings

Codes with parameters [55,5,33] and [77,7,44] were constructed.

Extended quadratic residue codes over Z_{121} have large permutation automorphism groups.

Permutation decoding is feasible due to the automorphism group structure.

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 $[77,7,44].$