Optimal STBCs from Codes over Galois Rings
Kiran T. and B. Sundar Rajan*
Dept. of ECE, Indian Institute of Science.
Bangalore-560022, INDIA.
Email: { kirant ,bsrajan}@ece.iisc.ernet.in
ABSTRACT
A Space-Time Block Code (STBC) CST is a finite collection of ntt x 1 complex matrices. If S is a complex signal set,
then CST is said to be completely over S if all the entries
of each of the codeword matrices are restricted to S. The
transmit diversity gain of such a code is equal to the minimum of the ranks of the difference matrices (Xfor
any X # X' E CST, and the rate is R = ' O g ' s l / c s T '
complex
symbols per channel use, where ICSTI denotes the cardinality of CST. For a STBC completely over S achieving transmit diversity gain equd to d, the rate is upper-bounded as
R 5 nt - d + 1. An STBC which achieves equality in this
tradeoff is said to be optimal. A Rank-Distance (RI)) code
CFF is a h e a r code over a finite field F,, where each codeword is a nt x E matrix over F,. RD codes have found applications as STBCs by using suitable rank-preserving maps from
F, to S. In this paper, we generalize these rank-preserving
maps, leading to generalized constructions of STBCs fiom
codes over Galois ring GR(p*,k). To be precise, for any
given value of d, we construct nt x 1 matrices over GR(pG,
k)
and use a rank-preserving map that yields optimal STBCs
with transmit diversity gain equal to d. Galois ring includes
the finite field Fp'i when a = 1 and the integer ring 2
,
. when
k = 1. Our construction includes as a special case, the earlier construction by Lusina et. al. which is applicable only for
FtD codes over F, ( p = 4s 1) and transmit diversity gain
d = nt.
x'),
+
1 INTRODUCTION
A quasi-static Raylejgh fading Multiple-input MultipleOutput (MIMO) channel with nt transmit and n, receive
antennas is modeled as
Yn,xI = H n , . x n t X ? b t x I
+Wn,xl,
unit variance. The collection of all possible transmit codewords X n r x lforms a Space-Time Black Code (STBC) CST.
From the pair-wise error probability point of view, it is
well-known that the performance of a space-time code at
high SNR is dependent on two parameters: transmit diversity gain and coding gain. The transmit diversity gain of
CST is the minimum of the ranks of the difference matrices
( X n txl Xklx l ) , for any X,, I # Xkt E CST,also called
the rank of CST. A nt x 1 STBC is said to be of full-rank
if it achieves the maximum transmit diversity nt (assuming
-
nt
51).
Let S denote a complex signal set (constellation). The
STBC CST is said to be completely over S if all the entries
of each of the codeword matrices are restricted to S [l].For
instance, the Alaniouti code {2] is completely over S if S is
chosen as the symmetric 6-PSK signal set. However, if S is
t h e symmetric 3-PSK signal set then this code is no longer
completely over S.
Following the convention in 13, 41, we define the rate of a
n, x I STBC completely over a signal set S as
R=
E
complex symbols/channel use,
where JCST~
denotes the cardinality of CST. If the code CST
achieves transmit diversity equal t o &, there exists a ratediversity tradeoff for space-time codes completely over S,
which is given by the relation
R 5 nt - & + 1.
(2)
A space-time code which achieves equality in the above
tradeoff is said to be optamal.
1.1 Space-time codes based on rank-distance codes
Several space-time code construction methods have been
where YnPxlis the received matrix over E channel uses,X,, xl recently investigated. A particularly interesting technique
involves STBC construction fiom rank-distance codes over
is the transmitted matrix, Hnpx,,
is the channel matrix and
finite
fields: A rank-distance (RD) code CFF (FF stands for
WnTxl
is the additive noise matrix, with the subscripts definite field) is a linear code over a finite field F,, where each
noting the dimension of the matrices, The matrices €Inrxx,,
codeword is a nt x l matrix over F,. For any pair of codewords
and WnTX1
have entries which are i.i.d, complex circularly
CI,CZ,the rank-distance between them, denoted as .,(Cl symmetric Gaussian random variabIes with zero mean and
C2),is defined to be the rank over F, of the nt x t difference
'This work was partly supported by the DRDO-IISc Program on
matrix C1 -Cz [5]. The rank of CFF,denoted by dq is defined
Advanced Research in Mathematical Engineering and by the Council
as the minimum of T ~ ( C Ca>
~ over all possible pairs of
of Scientific & Industriat. Research (CSIR), India, through Research
Grant (22(0365)/04/EMR-II) to B.S. Rajan.
distinct codewords. A standard method of constructing RD
0-7803-8964-61051520.00 0
-
IEEE
ICPWC'2005
code is to start with a length nt linear code over F,[ and
then map each codeword to a 1 x nt matrix by expanding
each entry of the codeword to a 2 x nt vector with respect
to a basis of F,t over F,. If k = log,l ICFF(?then using the
Singleton bound on C F F , it is possible to show that
k
- <nt-d,+I,
(3)
1 -
to a signal set that is a subset of some appropriate number
field. We construct optimal STBCs for any given value of
transmit diversity gain d 5 nt <_ 1. Our ccnstruction specializes to the technique'used in [IO] for a = f = I and
p = 4R 1, and our results also prove that the map used in
1101 is not only full-rank preserving (d = nt) but also preserves any arbitrary rank (d < nt). The contributions of this
paper are:
+
where dq is the rank of the code CFF. This bound is similar
to (Z), where we need to replace R with k / l and & with d,.
A rank d, code achieving this bound with equality is said to
be a rank-d, maximat rankdistance (MRD) code. If d, = nt,
then CFF is just a full-rank R;D code if k < E and a full-rank
MRD code if k = 1.
Application of RD codes to space-time code construction
was first proposed by Hammons and El G a m d in [6]. Fullrank space-time codes completely over S (a BPSK or QPSK
constellation) were obtained starting from a binary full-rank
In subsequent works
code, via a map from Fitx ' to SntX1.
14,7,8, 91, Lu and Kurnar have generalized this construction
b the so cdled Generalized Unified (GU) construction, with
a map from F;t x E to Sntxl,where p is any prime (including
2) and S is a subset of the ring Z[W,K];W,K is a primitive
pKth root of unity. In a related work [lo]?full-rank MRD
codes over JFp where p is of the form 4k 1 have been used
for constructing space-time codes, which involves:
STBCs with optimal rate-diversity tradeoff are constructed using codes over arbitrary Galois rings, which
generalizes the map in [lo].
Galois ring GR(pa,f) has been used €or constsucting conventional error-correcting codes and in this context, structure of linear codes over Galois ring and Singleton bound for such codes is very well understood
112, 131. In this paper, we exploit this structure to
show that the rank of STBC constructed from this
code over GR(p",f) is never less than the rank of a
unique RD code over F,j . This is the first work which
uses codes over arbitrary Galois rings for constructing
STBCs, apart from {4],where only GR(2',,) is used
for constructing STBCs completely over the QPSK signal set.
+
Constructing CFF;a nt x nt full-rank MRD code over a
field F,, p = 4k i1, and
Using a one-one map q5 from F, to a complex signal set
S to obtain a STBC,
CST = ( d ( C )
i c E CFF) >
where
STBC FROM CODE OVER GALOIS RING
Let K be a number field (a finite degree extension field of
the field of rational fractions 0) and & denote the ring of
integers in K,which is the set of all dements in TK which are
roots of some polynomial over Z.Let !$ be a prime &a1 in
%E containing a unique integes prime number p E Z. It is
well-known that the factor group Z K / is
~ isomorphic to a
finite field with characteristic p. Let the size of this finite field
be p f ( 9 l P ) . We call f ( v / p )as the inertial degree of over p.
Our main principle for constructing ogtimal STBCs hinges
on the isomorphic map from the Galois ring GRW, f[plp))
to Z&F, given by the following theorem.
Theorem 1 Let p be the unique rbtional integer prime in the
prime ideal 'p c &. The factor group &/ya as isomol-phic
to Q Galois ring GR@, f ) , where f = f (Q3jp) as the inertial
degree o f y over p .
Fzlrthemore, if !j?= ( T ) is a psilacipal ideal and 'RFF
i s a complete coset representative set for &f Q, then every
element in ZKhas a unique T-adiF representation:
In this paper, we generalize the results in 110) by working
in a more general setting; we construct STBCs €tom codes
over Gdois ring GR(p",f). A Galois ring GR(j9,f ) is a
ring isomorphic t o the residue class ring Z,n/(g(s)), where
Z,. is the integer ring modulo'p and f(x) i s a monic basic
irreducible polynomial over Z,. , of degree f Ill].It is local
ring with pGR@",f ) being the unique maximal ideal. The
quotient GR(pa,f)jpGRW, f ) is isomorphic to F,j and we
use the notation i to denote the image of 3: E G R ( p a , f )in
GRW, f)lPGR(PR,
fb
In this paper, we use nt x E codes over Gdois ring
GR(pa,f ) and give a rank preserving map from GR(p",f )
with rj E RFF,and the set
is
121
Q
complete coset representative set f o ~
z{i] = {U f ib 1 a,b E E ) .
2.1 Main principle for STBC construction
Suppose CCR is a nt x 1 code over a Galoia ring GRW, f)
and we want to construct a nt x 1 STBC using CGR. kt S
be equaI t o 8 compiete representative set of the quotient
ring &/!Jl".
This means, S = { T I , T Z , .... r p 9 f }such that
no two elements in S belong to the same coset in &/Va,
i.e., the Coset TI 33" # Tk $for an)' T j # T k E s, and
va
+
(rj
+ 'pa)-I( ~ +k Ips)
(Tj
It can be shown that the ideal
genemted b y R = (2 -t i) is a prime ideal in Z [ i ] and this
MkaI contains p = 5 . The set RFF= { O , f l , f i ) can be
chosen ns a complete mpresentata'ue set for
(n), usang
which, .rr-adic representation for few elements in Z[i]is given
below.
+ v a ) ( r k+'pa)
=
( ~ l +'pa),
=
(rm +%a),
€or some rt,7;n E S that are unique for the pair r j , r k . Alternately, we use the notation r j f T k mod Qat o mean T j
and rk belong to different cosets, and we extend the same
convention to the above two operations:
r3 + r k
rjrk
G , ~1
=
pa.
We call these operations as "modulo pa''addition and multiplication on S. Thus, the set S is a Galois ring with respect
to "modulo ya"addition and multiplication.
We use the set S as a signal set for constructing STBC
from CCR. Let 4 denote the isomorphism from GR(pa,f) to
&/'pa and 4-l denote the inverse map. The isomorphism
4-l restricted to the set S, is a one-one map from S to
GR(pa,f). The space-time code C s over S is then obtained
+
+
-
.........
.
Remark 1 The set 7 2 i s~one~of the many signal sets isomorphic to the Galois ring GRtp",f ) . This need not be the
one with minimum avemge energy.
Example 2 Let i = &
undK
i = Q ( i ) = { a + i b 1 a,b E
Q } . The ring of algebraic integers of this number field i s
122
-2
+ air
+
i -+ (i>n (-i>7r2
-i + 'C-i).
4- (i).
2
.
...... ......
......
.....
.... ......
.
..
..
..
--...*s .... .........i..
..!..
.,*!? ..... j ..... .....
:
:
.
..
.
..
:
..:.: .....!:..
.. .:.
'
.
.
.....
.... ........
Figure 2: Signal sets isomorphic to GR(52,1).
Example 3 Let IK = Q(i) and p be a prime of the form
4k + 3 (for e.g., p = 3). It is knoum that the ideal generated
by such p is a prime ideal in Z[i]and Z[i]/ (p) is isomorphic
to the finite field F P z . The set RFF= { a
ib 1
5
a,b 5
is a complete representative set for Z[il/ (p) and
RCR= (a +ib I
5 a,b 5
s h o w in Figure 3 is
a complete representattve set for the Gulois ring GR@",2).
9)
Figure 1: Signal set isomorphic to Z/pnZ.
13- (-1)r
i + (-i)7r
The corresponding set RGRshown in Figure 2(a) i s a complete representative set f o r the Galois ring GR(5*,1 ) . A s
mentioned eurlaeker, this need not be the signal set with mint
mum energy. I n Figure 2(b) we show another signal set isomorphic to GR(S2,1 ) with lesser average energy than that
of Figure .!?(a).
.......
Example 1 When K = Q and = p Z , the set RFFcan be
chosen as { O , H , ....
For t M s special case, the above
theorem states that every element of Z can be expcpressed in
the form ro f r l p rzp2 f . , rj E R F Fwhich
,
i s the wellknown p-adic represcnzation of any integer. Further, the set
RGR= (0, kl,. .~,
shown in Figure 1 is isomorphic
to z / p a z .
-l+T
-1i-i
3
-3
as
CsT = {9(C) I C E CGR)
where 4(C) is as given in (4). The size of the STBC so
obtained is equal to the size of the code CGR.
The Galois ring GR(pa,j )is a finite field when a = 1, in
which case, this construction reduces to the STBC construction from RD codes over arbitrary finite fields 114,151,
i 4"(-1)T
1-2
mod pa,
rm mod
-i+n
2
-2
l f i
-1-i
+
-v
--?
w)
The above examples illustrated construction of signal sets
isomorphic t o Galois rings. Optimal STBC construction using these signal sets will be dealt in the following subsections. W e the rank metric for matrices over finite fields
and codes over finite fields with this metric has been a subject of interest among coding theorists, we are not aware of
any such work on codes over finite rings. ln this paper, we
exploit the well-known structure of linear codes over Galois
.
*
. .
*"*
over B by the map C = [CO,
Cl,. ...C,,,-l] f A"* going
. to
[CO,
Cl,....C,,-1IT E B"? The corresponding tower of
codes for CGR is
''.
-
CCR = (cGR : PO) E (cGR : p') c . . c (CGR :pa-11,
+
..
..
. .i
................. i .... *. .....+!..,..
. .
where each of them is a nt x I RD code over KB.It is easy
to verify that (CCR : p') is the same code that is obtained by
mapping every vector of (C :p') to the corresponding nt x I
matrix.
...i
..................+...
Figure 3: Signal set isomorphic to GR(f,2).
rings [12,13].It is known that there exist a set of linear codes
over the residue field F,, of GR@a,f ) associated with every
linear code over GR(p",j ) . We relate the rank of the spacetime code constructed €rom codes over GR@, f ) to the rank
(over F,,) of the largest of these component codes, which is
the link for proving optimality of STBCs constructed in this
paper.
Theorem 2 Let CGR .be U nt x l linear code over GR(pa,f)
obtained as above and SGR be the signal set isomorphic
to GRW, f). The transmit diversity gain of the ne x I
STBC over SGR(obtained as in" .S
2.1) is q d to the
tmnsmit-diversity gain of the corresponding nt x 1 STBC obtained using the RD code (CGR : paA1).
Theorem 3 (Optimal STBCs) Let p be Q rational prime,
a, j be positive rational integers and R = nt - d + 1, f o r
1 5 nt 5 1 < 00; 1 5 d 5 nt. Let GRIP",fi) be an eztension
of the Galois ring GR(p4,I ) , with [ as a primitive element.
Consider the set of pdynomiat
6.8 Structure of codes over Galois rings [12, 131
A linear code of length nt WET A = GR(pa,f 1 ) is an Asubmodule in Ant, and any such linear code has a generator
matrix that is permutation equivalent to a matrix in the
standard from
XOJ
...
...
X0,z
...
0
0
f-lIkn-*
=
[x,T
"" 1
PXl,a
pa-1
pXT
Xk-1 ,a
a . '
p+lX,T-JT,
where X ~ , and
C Xj are matrices over A and the columns in
the above generator matrix are grouped into blocks of size
ko,kl , . . , , k , - 1 . The size of C is plr, where T =
k j ( aj ) , and the code C is said to be of type (k,, kl,...,k a - l ) .
For any x E A, if !t denotes the projection map on to the
residue field KA = Fp,t and if (C: z)= {e €,Ant:z e E C),
then there e x i s t a tower of linear codes over KA
x;ci
?I
such
=
that
- -
(e : PO) g (C : p i ) c . f . E (C :p - l }
(C : p 3 )
T
-T
... xj 1 .
and associate with every polynomial g(x) E 9, a nt x 1 matriz
where g( < j ) denotes the representation of g((jj) as a 1 x 1 vettor over GR(pn,f), using the ordered bcksis {I,<, ...,<'-'j.
Then the collection of p a f l R codeword matrices CCR =
IC, I g(2) E G} is o linear nt x 1 fTee code WeT GRW, f),
and the STBC CST = ~ ( C G R )is an optimal code over the
signal set SGR with transmit diversity gain equal to d.
Example 4 Let p = 3,a = 1 ond f = nt = 1 = d = 2.
For these panmeters, the base ring GR(pa,f) i s the finite
field F32 and the extension ring is the finite field Fp. The
maximal rate is then equal to R = nt d -k 1 = 1. We
use the primitive polynomial @ ( x ) = x4 + x
2 for constructing the field Fp. I f < is a root of @.(.), the subfield
F9 = {0,1,C10,<20
.....c7') und {1,<)is U basis for Fgl
-
+
OaeT Fq.
has
generator
matrix
The code C is said to be
In this ezample, the minimal polynomial of 5 O V ~ TFg is
- Cwx [lo (divides @e(x)). The code
+
@I(X) = x
4 = {PIS)
I
= (so is10x so7 SI E F9) I
and the corresponding m a t k representation i s
The ring A = GR@a,fZ) is a simple and separable extension of B.= GR(p",f),which means there exists an invertible element E A such that A = B(<)and the residue field
KA = KB(T);with K3 = F,,. Thus the set (1, ...,<'-'I
is a free-basis for A over B. We use g to denote the rep.resentation of IE A as a 1 x 1 coefficient vector over B
with respect to the ordered basis {I,[,. .. ,('-l}. If C is a
length nt linear code over A, we get; CGR: a nt x 1 linear code
<
c,
This code is an MRD code. Since p = 3 and f = 2, we can
RGR = ( m + in : -i 5 m,n 5 1) in
Ezample 3. The codewords of this STBC are of the form
use the signal set
123
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49,no. 10,pp. 2596-2616, Oct 2003.
100
[2] S.M. Alamouti, “A simple transmit diversity technique
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91
3 10’’
tP
d
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0 IO“
..................................
10-a
5
., . . . . . . . . . . . .
za
15
10
25
SNR (dB)
[4]H.-F. Lu and P. V. Kumar, “Ratediversity tradeoff of
Figure 4: Performance comparison for nt = E = 2 and n, = 1
space-time codes with fixed alphabet and optimal constructions for PSK modulation,” IEEE Trans. Infom.
Theory, vol. 49, no. 10, pp. 2747-2751, Oct 2003.
STBCs
1oa
[SI E. M. Gabidulin, “Theory of codes with maximum rank
distance,” Problemy Peredachi I n f o n a t s i i , vol. 21. no.
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...............; .........
10-1
10
..;.
[7] H.-F. Lu and P. V. Kumar, “Constructing optimal
space-time codes over various signal constellations,”,
in Pmc. IEEE Global Telecommunications Conf., San
Francisco, 1-5 Dec 2003, vol. 4, pp. pp.1957-1962.
............;...............;..............
I
25
20
15
30
35
SNR (dB)
Figure 5: Performance comparison for nt = 1 = 2 and n, = 1
[SI H.-F. Lu and P. V. Kumar, “A unified construction of
STBCs
where SO,SI E ( O , l ,
iuted below.
c2’, . . . .
<lo,
space-time codes with optimal rate-diversity tradeoff,”
submitted to IEEE Trans. Inform. Theory, 2003.
C70}
and the map
4 is tabu-
191 H.-F. Lu and P. V. Kumar, “Generalized unified construction of space-time codes with optimal rate-diversity
tradeoff,” in Pmc. Intl. Symp. on Inform. Theory,
Chicago, June 27-July 2 2004, p. 95:
[lo] P. Lusina, E. Gabidulin, and M. Bossert, “Maximum
rank distance codes as space-time codes,’’ IEEE %ns.
Inform. Theory, vol. 49, no. 10, pp. 2757-2760, Oct
In Figure 4 , we compare the performance of this code with
the full-rank optimal STBCs obtained by the GU constmctzon for nt = I = 2 and n, = 1. We consider two diferent
2003.
codes that QX constructible using GU construction: one corresponding to the parameters p = 3, K = 2,U = 1 uses the
9-PSKconstellation and the other corresponding to the pam m e t e r s p = 3,K = l , U = 2 , q = 3 and n = l. AlZ three
codes consist of g2 = 81 codewords. while the slope of the
codeword e m r probability is the same foor all codes, our code
has better coding gain.
[I11 McDonald B.R, Finite Rings with Identity, Marcel
Dekker Inc., New York, 1974.
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[13] G . H.Norton and A. Salagean, “Onthe Hamming distance of linear codes over finite chain rings,”, IEEE
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In Figure 5, we compare the performance of STBC over
the signal set in Figure 2(b) (constructed from Galois ring
2/s22) and the corresponding GU constructed code with
[141 Kiran T. and B. S. Rajan, “0ptimal.rate-diversity
tradeoff STBCs from codes over arbitrary Enite fields,”
subinitted t o the IEEE Intl. Conf. on Comm. (ICC
2005), Korea, Aug 2004.
same rate and diversity for nt = 1 = d = 2 and n, = 1.
Clearly, our code with better coding gain outperforms the
GU constructed cods.
1151 Kiran T. and B. S. Rajan, “STBCswith optimal ratediversity tradeoff from codes over Galois rings,” sub
mitted t o IEEE Trans. Inform. Theory, Oct 2004.
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