Abstract
This paper gives lower and upper bounds on the covering radius of codesover $\mathbb{Z}_{2^s}$ with respect to homogenous distance. We also determine the covering radius of variousRepetition codes, Simplex codes (Type $\alpha$ and Type $\beta$)and their dual and give bounds on the covering radii for MacDonald codes of both types over $\mathbb{Z}_4$.
Highlights
There has been a burst of activities and research in codes over finite rings in last decade, In particular codes over Zps and Z4 received much attention [1, 3, 4, 5, 10, 6, 11, 13, 14, 9, 12, 18]
We investigate the covering radius of the Z4 simplex codes and their duals, MacDonald codes and repetition codes
Since for the codes over Z2s various distances are possible we give a definition of the covering radius for a general distance which could be any of the possible distance
Summary
There has been a burst of activities and research in codes over finite rings in last decade, In particular codes over Zps and Z4 received much attention [1, 3, 4, 5, 10, 6, 11, 13, 14, 9, 12, 18]. The covering radius of codes over Z4 has been investigated with respect to Lee and Euclidean distances [16]. We investigate the covering radius of the Z4 simplex codes (both types) and their duals, MacDonald codes and repetition codes. The Hamming, Homogeneous / Lee and Euclidean distances dH(x, y), dHW (x, y)/dL(x, y) and dE(x, y) between two vectors x and y are wH(x − y), wHW (x − y)/wL(x − y) and wE(x − y), respectively. In this paper we define the covering radius of codes over Z2s with respect to different distances and in particular study the covering radius of Z4-simplex codes of type α and β namely, Skα and Skβ and their duals, MacDonald codes and repetition codes.
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