On Normal -Ary Codes in Rosenbloom-Tsfasman Metric

The notion of normality of codes in Hamming metric is extended to the codes in Rosenbloom-Tsfasman metric (RT-metric, in short). Using concepts of partition number and -cell of codes in RT-metric, we establish results on covering radius and normality of -ary codes in this metric. We also examine the acceptability of various coordinate positions of -ary codes in this metric. And thus, by exploring the feasibility of applying amalgamated direct sum method for construction of codes, we analyze the significance of normality in RT-metric. 1. Introduction Covering properties of codes have unique significance in coding theory, and covering radius, one of the four fundamental parameters, of a code is important in several respects [1]. Considering the fact that it is a geometric property of codes that characterizes maximal error correcting capability in the case of minimum distance decoding, covering radius had been extensively studied by many researchers (see, e.g., [2, 3] and the literature therein) especially with respect to the conventional Hamming metric. In fact, it has evolved into a subject in its own right mainly because of its practical applicability in areas such as data compression, testing, and write-once memories and also because of the mathematical beauty that it possesses. More on covering radius can be found in the monograph compiled by Cohen et al. [4]. In order to improve upon the bounds on covering radius, various construction techniques that use two or more known codes to construct a new code were proposed over the last few decades. One such method is direct sum construction which is a basic yet useful construction method. To improve upon the bounds related to covering radius of codes obtained using this method, the notion of normality which facilitates a construction technique known as amalgamated direct sum (ADS) was introduced for binary linear codes by Graham and Sloane in [5]. The same concepts were extended to binary nonlinear codes by Cohen et al. in [6]. Later, Lobstein and van Wee generalized these results to -ary codes [7]. In the present paper, we extend the notion of normality to codes in Rosenbloom-Tsfasman metric (RT-metric, in short). The present work is a generalization of our work on binary codes [8] to -ary case (though the discussed results hold good for codes over any alphabet of size , for simplicity, we have taken it to be the finite field ). RT-metric was introduced by Rosenbloom and Tsfasman in [9] and independently by Skriganov in [10] and is more adequate than Hamming metric in dealing with channels in which errors