research article on normal -ary codes in rosenbloom...
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Research ArticleOn Normal ๐-Ary Codes in Rosenbloom-Tsfasman Metric
R. S. Selvaraj and Venkatrajam Marka
Department of Mathematics, National Institute of Technology Warangal, Andhra Pradesh 506004, India
Correspondence should be addressed to R. S. Selvaraj; [email protected]
Received 10 February 2014; Accepted 17 March 2014; Published 2 April 2014
Academic Editors: A. Cossidente, E. Manstavicius, and S. Richter
Copyright ยฉ 2014 R. S. Selvaraj and V. Marka. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
The notion of normality of codes in Hamming metric is extended to the codes in Rosenbloom-Tsfasman metric (RT-metric, inshort). Using concepts of partition number and ๐-cell of codes in RT-metric, we establish results on covering radius and normalityof ๐-ary codes in this metric. We also examine the acceptability of various coordinate positions of ๐-ary codes in this metric. Andthus, by exploring the feasibility of applying amalgamated direct summethod for construction of codes, we analyze the significanceof normality in RT-metric.
1. Introduction
Covering properties of codes have unique significance incoding theory, and covering radius, one of the four funda-mental parameters, of a code is important in several respects[1]. Considering the fact that it is a geometric property ofcodes that characterizes maximal error correcting capabilityin the case of minimum distance decoding, covering radiushad been extensively studied by many researchers (see, e.g.,[2, 3] and the literature therein) especially with respect tothe conventional Hamming metric. In fact, it has evolvedinto a subject in its own right mainly because of its practicalapplicability in areas such as data compression, testing, andwrite-once memories and also because of the mathematicalbeauty that it possesses. More on covering radius can befound 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 knowncodes to construct a new code were proposed over the lastfew decades. One such method is direct sum constructionwhich is a basic yet useful construction method. To improveupon the bounds related to covering radius of codes obtainedusing this method, the notion of normality which facilitatesa construction technique known as amalgamated direct sum(ADS)was introduced for binary linear codes byGraham andSloane in [5]. The same concepts were extended to binary
nonlinear codes by Cohen et al. in [6]. Later, Lobstein andvan Wee generalized these results to ๐-ary codes [7].
In the present paper, we extend the notion of normalityto codes in Rosenbloom-Tsfasman metric (RT-metric, inshort). The present work is a generalization of our workon binary codes [8] to ๐-ary case (though the discussedresults hold good for codes over any alphabet of size ๐, forsimplicity, we have taken it to be the finite field F๐). RT-metric was introduced by Rosenbloom and Tsfasman in [9]and independently by Skriganov in [10] and is more adequatethan Hamming metric in dealing with channels in whicherrors have a tendency to occur with a periodic spikewiseperturbation of period ๐ โ N. As a generalization of theclassical Hammingmetric with richmathematical beauty andbeing advantageous over Hamming metric in dealing withcertain channels [9, 10], this metric attracted the attentionof coding theorists over the last two decades [11โ13]. Thecovering problem in RT-metric was first dealt with by Yildizet al. for RT-spaces over Galois Rings [14].
The organization of the present paper is as follows. InSection 2, some basic definitions and notations that are usedin this paper are presented. In Section 3, we have studied thecovering radius of ๐-ary codes in this metric by introducingtwo new tools, namely, partition number and ๐-cell whichgreatly reduce the difficulty in determining the coveringradius. In Section 4, we investigate the normality of ๐-ary
Hindawi Publishing CorporationISRN CombinatoricsVolume 2014, Article ID 237915, 5 pageshttp://dx.doi.org/10.1155/2014/237915
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codes in RT-metric. In this section, we also discuss thepossibilities of extending the direct sum and amalgamateddirect sum (ADS) constructions from Hamming metric toRT-metric. And, finally, Section 5 provides the conclusions ofthis paper.
2. Definitions and Notations
For ๐ฅ = (๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐ ) โ F๐
๐, where F๐ = GF(๐) is a finite field
of ๐ elements, we define the ๐-weight (RT-weight) of ๐ฅ to be
๐ค๐ก๐ (๐ฅ) = max {๐ | ๐ฅ๐ ฬธ= 0, 1 โค ๐ โค ๐ } . (1)
The RT-distance or ๐-distance between ๐ฅ = (๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐ )and ๐ฆ = (๐ฆ1, ๐ฆ2, . . . , ๐ฆ๐ ) โ F
๐
๐can be defined by ๐๐(๐ฅ, ๐ฆ) =
๐ค๐ก๐(๐ฅโ๐ฆ).The subsets of the space equippedwith thismetricare called RT-metric codes over F๐ (or ๐-ary RT-metric codes)and the subspaces are called linear RT-metric codes over F๐.The value ๐min = min{๐๐(๐ฅ, ๐ฆ) : ๐ฅ, ๐ฆ โ ๐ถ} is called theminimum ๐-distance of the code ๐ถ. The maximum distanceof any word in the ambient space from an RT-metric code iscalled the covering radius of the code. A code in which eachnonzero codeword is of the same weight ๐ค๐ is said to be aconstant weight code. A ๐-ary code of length ๐ , cardinality ๐พ,andminimum๐-distance๐๐ is said to be amaximumdistanceseparable code (an MDS code, in short) if๐พ = ๐(๐ โ๐๐+1).
Throughout this paper, unless otherwise specified, by acode we mean a ๐-ary RT-metric code and by distance wemean RT-distance. Moreover, [๐ , ๐, ๐๐]๐๐ denotes a ๐-arylinear code of length ๐ , dimension ๐, minimum distance ๐๐,and covering radius ๐ , (๐ , ๐พ, ๐๐)๐๐ denotes a ๐-ary code withcardinality ๐พ, and ๐ก๐[๐ , ๐] denotes the minimum coveringradius that a ๐-ary linear code of length ๐ and dimension ๐can possess.
Definition 1 (direct sumof two codes). Let๐ถ1 and๐ถ2 be codeswith parameters (๐ 1, ๐พ1)๐ and (๐ 2, ๐พ2)๐, respectively. Then,the direct sum of ๐ถ1 and ๐ถ2, denoted by ๐ถ1โจ๐ถ2, is definedas
๐ถ1โจ๐ถ2 = {(๐1, ๐2) | ๐1 โ ๐ถ1, ๐2 โ ๐ถ2} , (2)
and is a code of length ๐ 1 + ๐ 2 and cardinality๐พ1๐พ2.
Definition 2 (๐-ary normal codes). Let๐ถbe a ๐-aryRT-metriccode of length ๐ , cardinality ๐พ, and covering radius ๐ , and,also, for each ๐ โ F๐, let ๐ถ
(๐)
๐denote the set of codewords in
which the ๐th coordinate is ๐.Then, thenorm of๐ถwith respectto the ๐th coordinate is defined as
๐(๐)
= max๐ฅโF ๐ ๐
{
{
{
โ
๐โF๐
๐๐ (๐ฅ, ๐ถ(๐)
๐)}
}
}
, (3)
where
๐๐ (๐ฅ, ๐ถ(๐)
๐) = {
min {๐๐ (๐ฅ, ๐ฆ) | ๐ฆ โ ๐ถ(๐)
๐} if ๐ถ(๐)
๐ฬธ= 0
๐ if ๐ถ(๐)๐
= 0.(4)
Now, ๐ = min๐๐(๐) is called the norm of ๐ถ and the
coordinates ๐ for which ๐(๐) = ๐ are said to be acceptable.
Finally, a code is said to be normal if its norm satisfies ๐ โค๐๐ + ๐ โ 1. If the code ๐ถ is not clear from the context, we usethe notations๐(๐)(๐ถ) and๐(๐ถ).
3. Covering Radius of RT-Metric Codes over F๐
Definition 3 (partition number of a ๐-ary RT-metric code).Let ๐ถ be an (๐ , ๐พ, ๐๐)๐ code in RT-metric. The largestnonnegative integer ๐, for which each ๐-ary ๐-tuple can beassigned to at least one codeword whose last ๐ coordinates areactually that ๐-tuple, is called the partition number of the code๐ถ. The code with partition number ๐ can be partitioned into๐๐ parts, each of which has the property that all its membershave the same ๐-ary ๐-tuple as their last ๐ coordinates. A partobtained in the above fashion is called an ๐-cell of the code,if at least one field element is not present in the (๐ โ ๐)thcoordinate of its member codewords. As the definition ofpartition number suggests, a code with partition number ๐contains at least ๐๐ codewords.
If ๐ถ is an [๐ , ๐, ๐๐]๐ linear code, then each of the ๐๐ parts
does have ๐๐โ๐ codewords in it and so does each ๐-cell by itsdefinition. If๐ถ1 is an ๐-cell of the linear code๐ถ, then ๐ฅ+๐ถ1 for๐ฅ โ ๐ถ\๐ถ1 will also be an ๐-cell different from๐ถ1.Thus, for lin-ear codes, if there is one ๐-cell, then therewill be ๐๐ such ๐-cells.
Now, we will observe that covering radius of a code canbe determined using the concepts of partition number and๐-cell. The definition of partition number serves as a tool indetermining the covering radius of an RT-metric code, asshown by the following theorem.
Theorem 4. Let ๐ถ be an (๐ , ๐พ, ๐๐)๐๐ code in RT-metric. Then,the partition number of ๐ถ is ๐ if and only if its covering radiusis ๐ โ ๐.
Proof. First, let the partition number of the code be ๐.Partition the code such that codewords in each part containa unique ๐-ary (๐ + 1)-tuple as their last ๐ + 1 coordinates.Associate each part with the respective (๐+1)-tuple. Since thepartition number of the code is ๐, the number of such partswill be less than ๐๐+1. Choose an ๐ฅ โ F ๐
๐, whose last ๐ + 1
coordinates constitute the (๐ + 1)-tuple that does not have apart associated with it. Such an ๐ฅ will be at distance ๐ โ ๐from the code, which is the maximum distance that a wordcan actually be from the code ๐ถ. Hence, the covering radiusis ๐ โ ๐.
Conversely, let the covering radius be ๐ = ๐ โ ๐. By thedefinition of covering radius, any word must be at distance atmost ๐ โ ๐ from the code, which means that, for each word,there is at least one codeword which agrees with the word inthe last ๐ โ (๐ โ ๐) = ๐ coordinate positions. This implies thatthe partition number of the code is greater than or equal to ๐.Let us assume that the partition number of the code is ๐ >๐. Then, to each ๐-ary ๐-tuple, we can associate a codewordwhose last ๐ coordinates coincide with that ๐-tuple. Thus,eachword in F ๐
๐will be at distance atmost ๐ โ๐, contradicting
the fact that covering radius of ๐ถ is ๐ โ ๐. Hence, the partitionnumber of the code ๐ถ is ๐.
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Now, the above theorem can be restated in terms of ๐-cell in the following manner, as the definition of the ๐-cellsuggests.
Corollary 5. Let ๐ถ be an RT-metric code of length ๐ over F๐.Then, the covering radius of ๐ถ is ๐ โ ๐ if and only if โ an ๐-cellof the code ๐ถ.
Proposition 6. Let ๐ถ1 and ๐ถ2 be any two RT-metric codesover F๐ with parameters (๐ 1, ๐พ1, ๐1)๐๐ 1 and (๐ 2, ๐พ2, ๐2)๐๐ 2,respectively. Then, their direct sum ๐ถ = ๐ถ1โจ๐ถ2 is an (๐ 1 +๐ 2, ๐พ1๐พ2, ๐)๐๐ code with minimum distance ๐ = ๐1 andcovering radius ๐ = ๐ 1 + ๐ 2, provided ๐ถ2 ฬธ= F ๐ 2๐ .
Proof. The direct sum of ๐ถ1 and ๐ถ2 is given by
๐ถ1โจ๐ถ2 = {(๐1, ๐2) | ๐1 โ ๐ถ1, ๐2 โ ๐ถ2} . (5)
Clearly, this is a code of length ๐ 1 + ๐ 2 and cardinality ๐พ1๐พ2.Now, since the minimum distance of ๐ถ1 is ๐1, there exist twocodewords ๐1 and ๐
1in ๐ถ1 such that ๐๐(๐1, ๐
1) = ๐1. Then,
the codewords ๐ = (๐1, ๐2) and ๐= (๐
1, ๐2) for some ๐2 โ ๐ถ2
will also be at distance ๐1 which is minimum among all thecodewords of ๐ถ. FromTheorem 4 and Definition 3, it is clearthat the covering radius of a code in RT-metric depends onthe partition number of the code and that the process ofpartitioning starts with the right most coordinate. Therefore,byTheorem 4, unless๐ถ2 is the space F
๐ 2๐, the partition number
of ๐ถ must be equal to that of ๐ถ2 which is ๐ 2 โ ๐ 2 and, hence,the covering radius of ๐ถ is ๐ 1 + ๐ 2 โ (๐ 2 โ ๐ 2) = ๐ 1 + ๐ 2.
Remark 7. If ๐ถ2 = F ๐ 2๐ , then the covering radius of the code๐ถ = ๐ถ1โจ๐ถ2 is ๐ = ๐ 1.
4. Normality of Codes in RT-Metric
Proposition 8. If ๐ถ is an (๐ , ๐พ, ๐๐)๐๐ RT-metric code over F๐,then๐(๐ถ) โฅ ๐๐ .
Proof. The proof follows directly from the definition ofcovering radius and that of norm of a code.
Theorem9. Any ๐-ary RT-metric code of length ๐ and coveringradius ๐ is normal and all the coordinates are acceptable.
Proof. As ๐ = ๐ , the partition number is 0 which meansthat there exists an ๐ผ โ F๐ which is not present as the lastcoordinate of any codeword in ๐ถ. If ๐ฅ = (๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐ ) โ F
๐
๐
is such that ๐ฅ๐ = ๐ผ, then ๐๐(๐ฅ, ๐ถ(๐)
๐) = ๐ for each ๐ โ F๐
and for any ๐ โ {1, 2, . . . , ๐ }. Thus, ๐(๐) = ๐๐ < ๐๐ + ๐ โ 1 =๐๐ +(๐โ1). Hence, the code is normal and all the coordinatesare acceptable.
Lemma 10. Let ๐ถ be any RT-metric code with length ๐ andcovering radius ๐ < ๐ ; then
๐(๐)
= (๐ โ 1) ๐ + ๐ , (6)
for each ๐ โ {๐ + 1, ๐ + 2, . . . , ๐ }.
Proof. For a coordinate position ๐ โ {๐ + 1, ๐ + 2, . . . , ๐ }, thenorm is given by
๐(๐)
= max๐ฅโF ๐ ๐
{
{
{
โ
๐โF๐
๐๐ (๐ฅ, ๐ถ(๐)
๐)}
}
}
. (7)
As the covering radius of ๐ถ is ๐ , it has partition number๐ โ ๐ . So, the definition of partition number implies thateach codeword can be associated with a unique (๐ โ ๐ )-tuplewhich actually consists of the last (๐ โ ๐ ) coordinates of thatcodeword and, also, that there exists at least one (๐ โ ๐ + 1)-tuple which is not the same as the last ๐ โ ๐ + 1 coordinatesof any of the codewords. Now, when we partition the codeinto ๐ parts ๐ถ(๐)
๐, for ๐ โ F๐, we can also partition the set of
all (๐ โ ๐ )-tuples into ๐ parts corresponding to the associatedcodewords in ๐ถ(๐)
๐. If we choose a word ๐ฅ โ F ๐
๐whose last
๐ โ ๐ + 1 coordinates do not match with the last ๐ โ ๐ + 1coordinates of any of the codewords, then, for this word,โ๐โF๐
๐๐(๐ฅ, ๐ถ(๐)
๐) = (๐ โ 1)๐ + ๐ which is the maximum value
that a word can give. Hence, the proof holds.
Theorem 11. Let ๐ถ be any RT-metric code of length ๐ withcovering radius ๐ < ๐ . Then, the coordinates ๐ โ {๐ + 2, ๐ +3, . . . , ๐ } are not acceptable.
Proof. Theminimum norm is
๐ โค ๐(๐), โ๐ โ {1, 2, . . . , ๐ } . (8)
By Lemma 10, ๐(๐) = (๐ โ 1)๐ + ๐ , for all ๐ โฅ ๐ + 1, whichimplies
๐(๐ +1)
< ๐(๐ +2)
< ๐(๐ +3)
< โ โ โ < ๐(๐ ). (9)
From (8) and (9), one can conclude that
๐ < ๐(๐), โ๐ โ {๐ + 2, ๐ + 3, . . . , ๐ } . (10)
This completes the proof.
The above theorem is not sufficient to arrive at a decisionon the acceptability of the ๐ th coordinate, when the codehas covering radius ๐ = ๐ โ 1. This can be settled by thefollowing theorem which says that no ๐-ary linear RT-metriccode with dimension more than 1 and covering radius ๐ โ 1has acceptable last coordinate.
Theorem 12. Let ๐ถ be an [๐ , ๐, ๐๐]๐๐ linear RT-metric codeover F๐ with ๐ > 1 and ๐ = ๐ โ 1. Then, the last coordinate isnot acceptable.
Proof. By Lemma 10,
๐(๐ )
= (๐ โ 1) ๐ + ๐
= (๐ โ 1) ๐ + ๐ โ 1 = ๐๐ โ 1
= ๐ (๐ โ 1) + ๐ โ 1 = ๐๐ + (๐ โ 1)
(since ๐ = ๐ โ 1) .
(11)
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In order to prove this theorem, we must show that thereexists at least one coordinate ๐ for which ๐(๐) < ๐(๐ ). Asthe dimension of ๐ถ is ๐, there will be ๐ coordinate positions๐1, ๐2, . . . , ๐๐ such that ๐ถ
(๐๐ก)
๐ฬธ= 0 for all ๐ โ F๐ and for each
๐ก = 1, 2, . . . , ๐. One such coordinate position is ๐ (but not ๐ โ1,as the partition number is 1). As ๐ > 1, there exists at leastonemore such coordinate position ๐ โ {๐1, ๐2, . . . , ๐๐} and ๐ ฬธ= ๐ wherein ๐ถ(๐)
๐ฬธ= 0 for all ๐ โ F๐, and so ๐
(๐)โค ๐(๐ โ 1) < ๐
(๐ ).This completes the proof.
But, when the code is either of dimension 1 or of constantweight, all the coordinates become acceptable as the followingresults suggest.
Proposition 13. Let ๐ถ be an [๐ , 1]๐ linear RT-metric code.Then, ๐ถ is normal and all the coordinates are acceptable.
Proof. It can easily be seen that the covering radius of ๐ถ iseither ๐ or ๐ โ1. If it is ๐ , the result follows byTheorem 9. If it is๐ โ1, there exists a 1-cell. Since๐ถ is of dimension 1, there exist๐ such 1-cells. By Lemma 10, ๐(๐ ) = ๐๐ โ 1. Now for each ๐ โ{1, 2, . . . , ๐ โ1}, either๐ถ(๐)
0= ๐ถ or |๐ถ(๐)
0| = 1. In the former case,
๐ถ(๐)
๐= 0 for all ๐ ฬธ= 0 and, in the latter case, |๐ถ(๐)
๐| = 1 for all ๐ โ
F๐. In any case,๐(๐)
= ๐๐ โ1.This proves the result stated.
Proposition 14. Let ๐ถ be a constant weight code of length๐ over F๐. Then, ๐ถ is normal and all the coordinates areacceptable.
Proof. It is easy to see that the covering radius ๐ of theconstant weight code ๐ถ is either ๐ or ๐ โ 1, as it can havepartition number of either 0 or 1. If๐ = ๐ , then, by Lemma 10,it is done. If ๐ = ๐ โ 1, then ๐ถ(๐ )
๐ฬธ= 0 for any ๐ โ F๐. Then, for
any ๐, 0 โ ๐ถ(๐)0
and |๐ถ(๐)0| โฅ 1. For an ๐ฅ โ F ๐
๐with ๐ฅ๐ = 0 and
๐ฅ๐ โ1 ฬธ= 0,๐(๐) equals ๐๐ โ 1. Hence, the result follows.
Remark 15. If ๐ถ = F ๐ ๐, then its partition number is ๐ and so its
covering radius ๐ is 0. By (3), for any ๐, ๐(๐) = (๐ โ 1)๐ + 0 =(๐ โ 1)๐. And so,๐(1) < ๐(2) < โ โ โ < ๐(๐ ) and๐(1) = ๐ โ 1 =๐๐ +๐โ1. Thus, the ambient space F ๐
๐is normal with the first
coordinate being the only acceptable coordinate.
4.1. Normality and Direct Sum Construction
Proposition 16. Let ๐ถ1 and ๐ถ2 be (๐ 1, ๐พ1, ๐1)๐๐ 1 and(๐ 2, ๐พ2, ๐2)๐๐ 2 RT-metric codes, respectively, such that ๐ 2 ฬธ= 0.Then,
(i) the direct sum ๐ถ = ๐ถ1โจ๐ถ2 is normal;(ii) all the coordinates corresponding to ๐ถ1 are acceptable
for ๐ถ;(iii) the coordinate ๐ for ๐ 1 + 1 โค ๐ โค ๐ 2 is accept-
able for ๐ถ only if the norm of ๐ถ2 with respect to thecoordinate ๐ โ ๐ 1 is equal to ๐๐ 2.
Proof. Let ๐ถ1 and ๐ถ2 be any (๐ 1, ๐พ1, ๐1)๐๐ 1 and (๐ 2,๐พ2, ๐2)๐๐ 2 ๐-ary RT-metric codes, respectively. Then, by
Proposition 6, their direct sum ๐ถ = ๐ถ1โจ๐ถ2 is an (๐ , ๐พ1๐พ2,๐1)๐๐ code with length ๐ = ๐ 1 + ๐ 2 and covering radius๐ = ๐ 1 + ๐ 2. Now, for ๐ โ {1, 2, . . . , ๐ 1}, by (3), we have
๐(๐)
(๐ถ) = ๐ (๐ 1 + ๐ 2) . (12)
Now, in order to determine the norm ๐(๐ถ), we have to find๐(๐)(๐ถ) for each ๐ โ {๐ 1 +1, ๐ 1 +2, . . . , ๐ 1 + ๐ 2}, the coordinate
positions corresponding to๐ถ2. But, we know by Proposition 8that๐(๐ถ2) โฅ ๐๐ 2 which implies๐
(๐)(๐ถ) โฅ ๐(๐ 1+๐ 2). Hence,
๐(๐ถ) = ๐(๐ 1+๐ 2) = ๐๐ < ๐๐ +๐โ1.Thus,๐ถ is normal and allthe coordinate positions corresponding to ๐ถ1 are acceptableand the coordinate position ๐ โ {๐ 1 + 1, ๐ 1 + 2, . . . , ๐ 1 + ๐ 2}corresponding to ๐ถ2 is acceptable if๐
(๐โ๐ 1)(๐ถ2) = ๐๐ 2.
4.2. Normality and Amalgamated Direct Sum Construction.The amalgamated direct sum of two codes is defined asfollows.
Definition 17. Let ๐ถ1 be an [๐ 1, ๐1]๐๐ 1 normal code with thelast coordinate being acceptable and let ๐ถ2 be an [๐ 2, ๐2]๐๐ 2normal code with the first coordinate being acceptable.Then,their amalgamated direct sum (or shortly ADS), denoted by๐ถ1โจฬ๐ถ2, is an [๐ 1 + ๐ 2 โ 1, ๐1 + ๐2 โ 1]๐(๐ 1 + ๐ 2 โ 1) code andis given by
๐ถ1ฬ
โจ๐ถ2 = {(๐1 | ๐ข | ๐2) : (๐1 | ๐ข) โ ๐ถ1, (๐ข | ๐2) โ ๐ถ2} .
(13)
Here, in this definition, we consider those codes ๐ถ1 and ๐ถ2which are normal, respectively, with the last coordinate andfirst coordinate being acceptable such that the parts ๐ถ(๐ )
1๐and
๐ถ(1)
2๐are nonempty for all ๐ โ F๐.
As Theorem 12 shows the nonexistence of linear codes ofdimension more than 1 whose last coordinate is acceptable,the significance of amalgamated direct sum and hence thatof normality in RT-metric are less as far as the resultingcovering radius is concerned. But, since the one-dimensionalcodes are normal and all the coordinates are acceptable, onecan only use this ADS construction method to combine a1-dimensional code with any other linear code, which maybe helpful in the construction of MDS codes as seen in thefollowing result.
Theorem 18. Let ๐ถ1 be an [๐ 1, 1]๐ MDS code with coveringradius ๐ก๐[๐ 1, 1] (which is normal with last coordinate beingacceptable) and let ๐ถ2 be an [๐ 2, ๐2]๐ MDS code with coveringradius ๐ก๐[๐ 2, ๐2]. Then, their amalgamated direct sum ๐ถ1โจฬ๐ถ2is an [๐ 1 + ๐ 2 โ 1, ๐2]๐ MDS code with covering radius ๐ก๐[๐ 1 +๐ 2 โ 1, ๐2].
Proof. From the definition of partition number and fromTheorem 4, it is easy to see that ๐ก๐[๐ , ๐] = ๐ โ ๐. So, ๐ก๐[๐ 1, 1] =๐ 1 โ 1 and ๐ก๐[๐ 2, ๐2] = ๐ 2 โ ๐2. Now, as the dimension of๐ถ1โจฬ๐ถ2 is ๐2 (which is less than or equal to ๐ 2), the coveringradius of๐ถ1โจฬ๐ถ2 depends only on the coordinates pertainingto the code ๐ถ2. But ๐ถ2 has covering radius ๐ 2 โ ๐2 and hence
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ISRN Combinatorics 5
has partition number ๐2. So, ๐ถ1โจฬ๐ถ2 also will have partitionnumber ๐2 and hence will have covering radius ๐ 1+๐ 2โ1โ๐2which is equal to ๐ก๐[๐ 1+๐ 2โ1, ๐2], which implies that๐ถ1โจฬ๐ถ2is MDS. This completes the proof.
5. Conclusion
We discussed the normality of codes over F๐ in RT-metricand determined its norm with respect to various coordinatepositions. We also established that the last coordinate is notacceptable for any nontrivial code in this metric whichmakesthe ADS construction less significant as far as the higherdimensional RT-metric codes are concerned.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
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