ISlT 2002, Lausanne, Switzerland,June 30 - July 5, 2002
f i l l Rank Distance Codes and Optimal STBC for BPSK Modulation
K. N. Manoj and B. Sundar Rajan
Dept. of Electrical Communication Engineering,
Indian Institute of Science, Bangalore,
Karnataka, India. PIN: 560 012.
kumminiQieee.org, bsrajanQece.iisc.ernet.in
Abstract
Viewing the codewords of an [n,k ] linear
code over a field Fqm as m x n matrices over Fq by expanding each entry of the codeword with respect to an
Fq-basis of Fqm, the rank weight of a codeword is the
rank over Fqof the corresponding matrix and the rank
of the code is the minimum rank weight among all
non-zero codewords. For m 2 n - k + l , codes with maximum possible rank distance, called Maximum Rank
Distance (MRD) codes have been studied previously.
In this paper, we study codes with maximum possible
rank distance for the cases m 5 n - k 1, calling them
Full Rank Distance (FRD) codes. Generator matrices
of FRD codes are characterized.
~
+
I. INTRODUCTION
Let Fq denote the Galois field with q elements and m and
n be positive integers. Let M:?, denote the set of all m x n
matrices over Fq. For any matrix M , we denote t h e rank of
M over Fq by rq( M ) .
Let f ( x ) = zm - fm-lzm--l - . . . - flz - f o be a primitive
polynomial over Fq. Let cy E Fqm be a root of f (x). T h e set
{ 1 , cy, c y 2 , . . . ,am-’} forms a polynomial basis for Fqm seen as
a vector space over Fq [ l ] .Therefore every element in Fqm
can b e seen as a polynomial in cy over Fq . T h e map
@ : Fg“-
+MkLn
is defined as follows: for any v =
E F,$ , @(v)is t h e
m x n matrix with t h e element at t h e i t h column and j t h row
being the coefficient of cuj in t h e polynomial expansion of vi.
Q, is an Fq -space isomorphism.
Let C b e an n-length code over Fqm i.e. C c Fpm . For
any pair of codewords c,c‘ E C, t h e rank distance between
them is defined to be r,(@(c- c’)) [2]. T h e rank dR,min of C
is given by:
dR,min := min rq(Q,(c
- c’))
Theorem 1.1. A n y n-length F R D code over Fqm has at most
qn elements.
C over Fqmis called a
Maximal Full Rank Distance (MFRD) code if (CI= 4 “ .
Definition 1. An n-length F R D code
11. CHARACTERIZATION
OF FRD CODES
Theorem 11.1. Let C be a n [n,k ] linear code over Fqm and G
be its generator matrix. Using the m a p Q, : F4m + M:?, ,
each of t h e m can be expanded t o matrices in
giving a
total of k m row vectors an FP . T h e n C is F R D if and only af
the k m vectors obtained as above are Fq-linear independent.
Theorem 11.2. An n-length M F R D code over Fqm exists
iff m divides n.
111. SPACE-TIME
BLOCKCODES FROM FRD CODES
A space-time block code (STBC) C, for a channel with m
transmit antennas and a quasi-static interval of n is a finite set
of m x n matrices with entries from a signal set S C C [ 5 ] . With
perfect CSI, t h e diversity of C is directly proportional t o t h e
minimum of ranks of pairwise differences of matrices in C over
t h e complex field C [ 5 ] . Optimal STBC for BPSK modulation
has been constructed as follows: let C be a n nm-length linear
code over Fz. Let (T be a parser t h a t maps vectors in C t o
m x n matrices over F2. Let C! b e t h e m x n S T B C obtained
by mapping matrices in a ( C ) using t h e m a p 0 ++ +1,1 I+ -1.
C achieves full spatial diversity m if and only if every nonzero
matrix in C is a matrix of full rank over FZ [6]. F R D codes
can b e used t o get STBCs t h a t achieve full diversity .
FRD codes can b e also used t o obtain FRD codes based on
Gaussian number fields [7].
C,C‘EC
cfc’
Since t h e rank distance between two codewords c, c’ E C is at
most t h e Hamming distance between them, we have,
dR,min
5 min{m, n - logq, IC1 4-
1)
+
Codes with n < m and dR,mi7L= n - log,, IC1 1 has been
studied in [2, 3 , 41 and are called MRD (Maximum Rank Dis= m and call
tance) codes. We consider codes with
them Full Rank Distance ( F R D ) codes.
Note 1. For a n y n-length F R D code over Fqm, m 5 n.
‘This work was partly supported by IISc-DRDO programme
on Advanced Research in Mathematical Engineering through a research grant to B. S. Rajan.
0-7803-7501
-7/02/$17.00@2002
IEEE.
276
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