Simplex and MacDonald Codes over $R_{q}$

In this paper, we introduce the homogeneous weight and homogeneous Gray map over the ring R q = F 2 [u 1 , u 2 ,. .. , u q ]/ u 2 i = 0, u i u j = u j u i for q ≥ 1. We also consider the construction of simplex and MacDonald codes of types α and β over this ring. Further, we study the properties of these codes such as their binary images and covering radius. 1 Introduction Codes over rings have been of significant research interest since the pioneering work of Ham-mons et al. [10] on codes over Z 4. Many of their results have been extended to finite chain rings such as Galois rings and rings of the form F 2 [u]/ u m. Recently, as a generalization of previous studies [14, 15], Dougherty et al. [5] considered codes over an infinite class of rings, denoted R q. These rings are finite and commutative, but are not finite chain rings. Motivated by the importance of the simplex and MacDonald codes which have been defined over several finite commutative rings [1,8,9], in this work, we define the homogeneous weight over R q and present simplex codes and MacDonald codes over this ring. The properties of these codes are studied, particularly the weight enumerators and covering radius. Further, the binary images of these codes are considered. The remainder of this paper is organized as follows. In Section 2, some preliminary results are given concerning the ring R q and codes over this ring. Further, we define the homogeneous weight and its Gray map. The simplex codes of type α and their properties and binary images are given in Section 3, while the simplex codes of type β and their properties and binary images are given in Section 4. In Section 5, the MacDonald codes of types α and β are presented along with their binary images. Section 6 presents the repetition codes and considers some properties of these codes, in particular the covering radius. Finally, in 1