In this talk, we will discuss about duadic codes (duadic group algebra codes) over some special class of finite rings. These codes are a family of abelian codes, which are themselves generalization of cyclic codes.

We will discuss about duadic codes of odd length over $\mathbb{Z}_4+u\mathbb{Z}_4, u^2=0$ and over $\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2, u^2=v^2=0, uv=vu$. We study these codes by considering them as a class of abelian codes and using the Fourier transform approach. In general, we will consider the algebraic structure of abelian codes over these rings. Some properties of the torsion and residue codes of abelian codes are studied. We will discuss about some results related to self-duality and self-orthogonality of duadic codes. Some conditions on the existence of self-dual augmented and extended codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ as well as over $\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$ will be determined.

We will also discuss about a new Gray map over $\mathbb{Z}_4+u\mathbb{Z}_4$ under which an abelian code over $\mathbb{Z}_4+u\mathbb{Z}_4$ is an abelian code over $\mathbb{Z}_4$. We have obtained five new linear codes of length $18$ over $\mathbb{Z}_4$ from duadic codes of length $9$ over $\mathbb{Z}_4+u\mathbb{Z}_4$ as images of Gray map and a new map defined from $\mathbb{Z}_4+u\mathbb{Z}_4$ to $\mathbb{Z}_4^2$. The parameters of these codes are $[18, 4^42^{10}, 4], [18, 4^52^8, 4], [18, 4^42^5, 8], [18, 4^02^9, 8]$ and $[18, 4^22^5, 6]$. The code with parameters $[18, 4^02^9, 8]$ is self-orthogonal.

We will then discuss about abelian codes over $\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$, and their Gray images.