On ℤ₄-Simplex Codes and Their Gray Images

Abstract

In [7]Rai ns has shown that for any linear code $\mathcal{C}$ over ℤ₄ d _H,the minimum Hammimg distance of $\mathcal{C}$ and d _L, the minimum Lee distance of $\mathcal{C}$ satisfy $d_H \geqslant \left\lceil {\tfrac{{^d L}} {2}} \right\rceil$. $\mathcal{C}$ is said to be of type α(β) if $d_H = \left\lceil {\tfrac{{^d L}} {2}} \right\rceil \left( {d_H > \left\lceil {\tfrac{{^d L}} {2}} \right\rceil } \right)$. In this paper we define Simplex codes of type α and β,namely, S ^α _k and S ^β _k , respectively over ℤ₄. Some fundamental properties like 2-dimension, Hamming and Lee weight distributions, weight hierarchy etc. are determined for these codes. It is shown that binary images of S ^α _k and S ^β _k by the Gray map give rise to some interesting binary codes.