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

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 $(c_0,c_1,\mathrm{.}\mathrm{.}\mathrm{.},c_{n-1})\in \mathcal{C}$ then the cyclic shift $(c_1,c_2,\mathrm{.}\mathrm{.}\mathrm{.},c_0)\in \mathcal{C}$. 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.