Rank Distance Bicodes and Their Generalization

2010·2.6 MB·English

by  Vasantha Kandasamy, W. B.

Rank Distance Bicodes - Cover:Layout 1 4/5/2010 1:41 PM Page 1
W. B. Vasantha Kandasamy
Florentin Smarandache
N. Suresh Babu
R.S. Selvaraj
RANK DISTANCE BICODES AND
THEIR GENERALIZATION
Svenska fysikarkivet
Stockholm, Sweden
2010
Svenska fysikarkivet (Swedish physics archive) is a publisher
registered with the Royal National Library of Sweden (Kungliga
biblioteket), Stockholm.
Postal address for correspondence:
Näsbydalsvägen 4/11, 18331 Täby, Sweden
Peer-Reviewers:
Prof. Ion Patrascu
Department of Mathematics
Fratii Buzesti College, Craiova, Romania
Prof. Catalin Barbu, Vasile Alecsandri College, Bacau, Romania
Dr. Fu Yuhua, 13-603, Liufangbeili
Liufang Street, Chaoyang district, Beijing, 100028,P. R. China
Copyright © Authors, 2010
Copyright © Publication by Svenska fysikarkivet, 2010
Copyright note: Electronic copying, print copying and distribution of
this book for non-commercial, academic or individual use can be made
by any user without permission or charge. Any part of this book being
cited or used howsoever in other publications must acknowledge this
publication. No part of this book may be reproduced in any form
whatsoever (including storage in any media) for commercial use
without the prior permission of the copyright holder. Requests for
permission to reproduce any part of this book for commercial use must
be addressed to the Author. The Author retains his rights to use this
book as a whole or any part of it in any other publications and in any
way he sees fit. This Copyright Agreement shall remain valid even if
the Author transfers copyright of the book to another party.
ISBN: 978-91-85917-12-9
2
CONTENTS
Preface 5
Chapter One
BASIC PROPERTIES OF
RANK DISTANCE CODES
7
Chapter Two
RANK DISTANCE BICODES
AND THEIR PROPERTIES
25
Chapter Three
RANK DISTANCE m-CODES
77
3
Chapter Four
APPLICATIONS OF RANK
DISTANCE m-CODES
131
FURTHER READING 133
INDEX 144
ABOUT THE AUTHORS 150
4
PREFACE
In this book the authors introduce the new notion of
rank distance bicodes and generalize this concept to Rank
Distance n-codes (RD n-codes), n, greater than or equal to
three. This definition leads to several classes of new RD
bicodes like semi circulant rank bicodes of type I and II,
semicyclic circulant rank bicode, circulant rank bicodes,
bidivisible bicode and so on. It is important to mention that
these new classes of codes will not only multitask
simultaneously but also they will be best suited to the
present computerised era. Apart from this, these codes are
best suited in cryptography.
This book has four chapters. In chapter one we just
recall the notion of RD codes, MRD codes, circulant rank
codes and constant rank codes and describe their
properties. In chapter two we introduce few new classes of
codes and study some of their properties. In this chapter
we introduce the notion of fuzzy RD codes and fuzzy RD
bicodes. Rank distance m-codes are introduced in chapter
three and the property of m-covering radius is analysed.
Chapter four indicates some applications of these new
classes of codes.
5
Our thanks are due to Dr. K. Kandasamy for proof-
reading this book. We also acknowledge our gratitude to
Kama and Meena for their help with corrections and
layout.
W.B.VASANTHA KANDASAMY
FLORENTIN SMARANDACHE
N. SURESH BABU
R.S.SELVARAJ
6
Chapter One
B P
ASIC ROPERTIES OF
R D C
ANK ISTANCE ODES
In this chapter we recall the basic definitions and properties of
Rank Distance codes (RD codes). The Rank Distance (RD)
codes are special type of codes endowed with rank metric
introduced by Gabidulin [24, 27]. The rank metric introduced by
Gabidulin is an ideal metric for it has the capability of handling
varied error patterns efficiently.
The significance of this new metric is that it recognizes
the linear dependence between different symbols of the
alphabet. Hence a code equipped with the rank metric detects
and corrects more error patterns compared to those codes with
other metric. In 1985 Gabidulin has studied a particular class of
codes equipped with rank metric called Maximum Rank
Distance (MRD) codes. Throughout this book Vn denotes a
linear space of dimension n over the Galois field GF(2N), N > 1.
By fixing a basis for Vn over GF(2N), we can represent any
element x (cid:2) Vn as an n-tuple (x , x , …, x ) where x (cid:2)(cid:3)GF(2N).
1 2 n i
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Again, GF(2N) can be considered as a linear space of dimension
‘N’ over GF(2). Hence an element x (cid:2)(cid:3)GF(2N) has a
i
representation as a N-tuple ((cid:4) , (cid:4) , …, (cid:4) ) over GF(2) with
i1 i2 in
respect to some fixed basis. Hence associated with each x (cid:2) Vn,
(n(cid:5) N) there is a matrix,
(cid:6)(cid:4) (cid:2) (cid:4) (cid:7)T
(cid:8) 11 1n (cid:9)
(cid:4) (cid:2) (cid:4)
(cid:8) (cid:9)
m(x) = 21 2n
(cid:8) (cid:3) (cid:3) (cid:9)
(cid:8) (cid:9)
(cid:10)(cid:4) (cid:2) (cid:4) (cid:11)
N1 Nn
where the ith column represents the ith coordinate ‘x’ of x over
i
GF(2). If we assume x(cid:3)(cid:2)(cid:3)GF(2N) has a representation as a N-
i
tuple x (cid:12)(cid:4) u (cid:13)(cid:4) u (cid:13)...(cid:13)(cid:4) u where u , u , …, u is some
i 1i 1 2i 2 Ni N 1 2 N
fixed basis of the field F regarded as a vector space over F . If
qN q
An denote the ensemble of all N (cid:14) n matrices over F and if
N q
A:Vn (cid:15)An is a bijection defined by the rule, for any vector x
N
= (x , …, x )(cid:2) Fn , the associated matrix denoted by,
1 n qN
(cid:6)a a (cid:2) a (cid:7)
(cid:8) 11 12 1n (cid:9)
A(cid:16)x(cid:17)(cid:12)(cid:8)a21 a22 (cid:2) a2n(cid:9)
(cid:8) (cid:3) (cid:3) (cid:3) (cid:9)
(cid:8) (cid:9)
(cid:10)a a (cid:2) a (cid:11)
N1 N2 Nn
where the ith column represents the ith coordinate x of x over F .
i q
Now we proceed on to define the rank of an elementx(cid:2)Vn.
DEFINITION 1.1:The rank of a vector x(cid:3)(cid:2)(cid:3)Fn is the rank of the
qN
associated matrix A(x). Let r(x) denote the rank of the vector x
(cid:2)(cid:3)Fn , over F .
qN q
By the usual properties of the rank of a matrix, it is easy to
prove the following inequalities.
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1. r(x)(cid:18) 0 for every x (cid:2) Vn.
2. r(x) = 0 if and only if x = 0.
3. r(x+y) (cid:5) r(x) + r(y) for every x, y (cid:2) Vn.
4. r(ax) = | a | r(x) for every a (cid:2) GF(2) and x (cid:2) Vn.
Thus the function x (cid:15) r(x) defines a norm on Vn, (x (cid:15) r(x)
defines a norm on Fn ) and is called the rank norm. In this book
qN
we denote the rank norm by r(x) or wt(x) or by || x ||. The rank
norm induces a metric called rank metric (or rank distance) on
Fn .
qN
DEFINITION 1.2: The metric induced by the rank norm is
defined as the rank metric on Vn(Fn ) and it is denoted by d .
qN R
The rank distance between x, y(cid:3)(cid:2)(cid:3)Vn is the rank of their
difference : d (x, y) = r(x – y). The vector space Vn (Fn ) over
R qN
F equipped with the rank metric d is defined as a rank
qN R
distance space.
DEFINITION 1.3: A linear space Vn over GF(2N), N > 1 of
dimension n such that n (cid:5) N, equipped with the rank metric is
defined as a rank space or rank distance space.
Now we proceed on to recall the definition of Rank Distance
(RD) codes.
DEFINITION 1.4: A Rank Distance code (RD-code) of length n
over GF(2N) is a subset of the rank space Vn over GF(2N). A
linear [n, k] RD code is a linear subspace of dimension k in the
rank space Vn. By C[n, k] we denote a linear [n, k] RD-code.
DEFINITION 1.5: A generator matrix of a linear [n, k] RD code
C is a k (cid:14) n matrix over GF(2N) whose rows form a basis for C.
A generator matrix G of a linear RD code C[n, k] can be
brought into the form G = [I , A ] where I is the identity
k k, n–k k
matrix and A is some matrix over GF(2n). This form is
k, n–k
called the standard form.
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