A family of Quadratic Resident Codes over $Z_{2^m}$

TL;DR

This paper introduces a new family of quadratic residue cyclic codes over rings of integers modulo 2^m, exploring their algebraic properties, self-orthogonality, and weight characteristics.

Contribution

It defines quadratic residue codes over Z_{2^m} using idempotent generators and analyzes their properties, extending classical binary quadratic residue code concepts.

Findings

Codes are self-orthogonal

Codes exhibit properties similar to binary quadratic residue codes

Discussion of Hamming weight characteristics

Abstract

A cyclic codes of length $n$ over the rings $Z_{2^{m}}$ of integer of modulo $2^{m}$ is a linear code with property that if the codeword $(c_0,c_1,...,c_{n-1})\in \mathcal{C}$ then the cyclic shift $(c_1,c_2,...,c_0)\in \mathcal{C}$. Quadratic residue codes are a particularly interesting family of cyclic codes. We define such family of codes in terms of their idempotent generators and show that these codes also have many good properties which are analogous in many respects to properties of binary quadratic residue codes. Such codes constructed are self-orthogonal. And we also discuss their hamming weight.