Simplex and MacDonald Codes over $R_{q}$

TL;DR

This paper introduces homogeneous weights and Gray maps over the ring $R_q$, and constructs simplex and MacDonald codes of types alpha and beta over this ring, expanding coding theory in algebraic structures.

Contribution

It presents new definitions of homogeneous weight and Gray map over $R_q$, and constructs new classes of simplex and MacDonald codes over this ring.

Findings

Defined homogeneous weight and Gray map over $R_q$

Constructed simplex and MacDonald codes over $R_q$

Extended coding theory to algebraic rings

Abstract

In this paper, we introduce the homogeneous weight and homogeneous Gray map over the ring $R_{q}=\mathbb{F}_{2}[u_{1},u_{2},\ldots,u_{q}]/\left\langle u_{i}^{2}=0,u_{i}u_{j}=u_{j}u_{i}\right\rangle$ for $q \geq 2$. We also consider the construction of simplex and MacDonald codes of types $\alpha$ and $\beta$ over this ring.