Optimal STBCs from Codes over Galois Rings - ePrints@IISc

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 collec- tion 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 min- imum of the ranks of the difference matrices

(X - x'),

^for

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 cardinal- ity of

CST.

For a STBC completely over S achieving trans- mit 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 code- word is a nt x E matrix over F,. RD codes have found applica- tions 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 G R ( 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 ear- lier construction by Lusina et. al. which is applicable only for FtD codes over

F,

( p = 4s

+

1) and transmit diversity gain d = nt.

1 INTRODUCTION

A quasi-static Raylejgh fading Multiple-input Multiple- Output (MIMO) channel with nt transmit and n, receive antennas is modeled as

Yn,xI = H n , . x n t X ? b t x I

+

W n , x l ,

where YnPxl is the received matrix over E channel uses, X,, xl

is the transmitted matrix, Hnpx,, is the channel matrix and WnTxl is the additive noise matrix, with the subscripts de-

noting the dimension of the matrices, The matrices €Inrxx,, and WnTX1 have entries which are i.i.d, complex circularly

symmetric Gaussian random variabIes with zero mean and 'This work was partly supported by the DRDO-IISc Program on Advanced Research in Mathematical Engineering and by the Council of Scientific & Industriat. Research (CSIR), India, through Research Grant (22(0365)/04/EMR-II) to B.S. Rajan.

unit variance. The collection of all possible transmit code- words X n r x l forms 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 diver- s i t y gain and coding gain. The transmit diversity gain of

CST

is the minimum of the ranks of the difference matrices 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 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

( X n t xl

-

^{Xkl x l ) , for any}

X,,

^I

#

^Xkt

nt

51).

E CST, _also called

complex symbols/channel use,

R =

E

where J C S T ~ denotes the cardinality of CST. If the code CST achieves transmit diversity equal t o

&,

there exists a rate- diversity tradeoff for space-time codes completely over

S,

which is given by the relation

R 5 nt

-

^&

+

^1.

A space-time code which achieves equality in the above tradeoff is said to be optamal.

1.1

Several space-time code construction methods have been recently investigated. A particularly interesting technique involves

STBC

construction fiom rank-distance codes over finite fields: A rank-distance (RD) code CFF (FF stands for finite field) is a linear code over a finite field

F,,

where each codeword is a nt x l matrix over F,. For any pair of codewords CI, CZ, the rank-distance between them, denoted as .,(Cl -

C2),

is defined to be the rank over

F,

of the ^nt x

t

difference matrix C1 -Cz [5]. The rank of

CFF,

denoted by dq is defined as the minimum of T ~ ( C ~

-

Ca> over all possible pairs of distinct codewords. A standard method of constructing RD

(2)

Space-time codes based on rank-distance codes

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

(3) k

1 -

-

< n t - d , + I ,

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, = n t , 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]. Full- rank space-time codes completely over S (a BPSK or QPSK constellation) were obtained starting from a binary full-rank code, via a map from Fit x ' to S n t X 1 . In subsequent works 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 S n t x l , where p i s 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:

Constructing C F F ; a nt x nt full-rank MRD code over a field

F,,

p = 4k i- 1, 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 C F F ) >}

where

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

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 ) GRW, f)lPGR(PR,

f b

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 spe- cializes to the technique'used in [IO] for a = f = I and p = 4R

+

1, and our results also prove that the m a p used in 1101 is not only full-rank preserving (d = nt) but also pre- serves any arbitrary rank (d

<

^nt). The contributions of this paper are:

STBCs with optimal rate-diversity tradeoff are con- structed using codes over arbitrary Galois rings, which generalizes the map in [lo].

Galois ring GR(pa,f) has been used €or constsuct- ing conventional error-correcting codes and in this con- text, structure of linear codes over Galois ring and Sin- gleton 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 sig- nal set.

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

ZK

has a unique T-adiF representation:

with rj E R F F , and the set

i s Q complete coset representative set f o ~

2.1

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,

Main principle for STBC construction

i.e., the Coset TI

+ 33" #

^{Tk $-}

va

^{for an)' T j}

#

^{T k E}

s,

^and

(rj

+

^{'pa) -I ( ~ k}

+

^{Ips) = ( ~ l + 'pa),}

( T j

+

v a ) ( r k +'pa) = ( r m ^+%a),

€or some rt,7;n E S that are unique for the pair r j , r k . Al- ternately, we use the notation r j f ^{T k} mod

Qa

t o mean T j

and rk belong to different cosets, and we extend the same convention to the above two operations:

r3 + r k G , ~1 mod

pa,

rjrk

=

rm mod

pa.

We call these operations as "modulo

pa''

^{addition and mul-}

tiplication 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

CsT = {9(C)

I

C E CGR) as

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

construc- tion from RD codes over arbitrary finite fields 114, 151, Example 1 When K = Q and = p Z , the set RFF can 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 F , which i s the well- known p-adic represcnzation of any integer. Further, the set

RGR = (0, kl,

.

^.

~,

shown in Figure 1 is isomorphic t o z / p a z .

Figure

.

1: Signal set isomorphic to Z/pnZ.

Remark 1 The set 7 2i s ~one ~of the m a n y signal sets iso- morphic to the Galois ring GRtp", f ) . This need not be the one with minimum avemge energy.

Example 2 Let i =

&i

u n d K = Q ( i ) = { a + i b

1

a,b E Q } . The ring of algebraic integers of this number field ^{i s}

z{i]

^{= {U f ib}

1

a , b E E ) . It can be shown that the ideal genemted b y R = (2 -t i) is _a prime ideal in Z [ i ] and t h i s 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.

2 -2 l f i -1-i

1 - 2 -1i-i 3 -3

-i+n i 4" (-1)T

1 3 - (-1)r i

+

^(-i)7r

i

-+

^(i>n

+

(-i>7r2 -i

+

^{'C-i). 4- (i)..}

- l + T

-2

+

^{a i r}

2

The corresponding set RGR shown in Figure 2(a) i s a com- plete 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 iso- morphic t o GR(S2, 1 ) with lesser average energy than that of Figure .!?(a).

... ... ...

... .... ...

... ... --...* 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 9)

^{is a} complete representative set for Z[il/ (p) and RCR = (a + i b

I -v

^{5 a , b 5}

w)

s h o w in Figure 3 is a complete representattve set for the Gulois ring

GR@",

2).

The above examples illustrated construction of signal sets isomorphic t o Galois rings. Optimal

STBC

construction us- ing these signal sets will be dealt in the following subsec- tions. W e the rank metric for matrices over finite fields and codes over finite fields with this metric has been a sub- ject 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

*

.

^*"*

. .

. .

. .

.

.i

... i .... *. ... +!...,...

...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 space- time 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.

6.8

A linear code of length nt WET A = GR(pa, f 1 ) is an A- submodule in Ant, and any such linear code has a generator matrix that is permutation equivalent to a matrix in the standard from

Structure of codes over Galois rings [12, 131

. . .

... PXl,a

... ""

^{Xk-1 ,a}

1

XOJ X0,z

0 0 f - l I k n - * pa-1

=

[x,T

pXT ^{a . '} p+lX,T-JT, where X ~ , C and 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 =

x;ci

^{k j ( a}

-

For any

x

E A, if !t denotes the projection map on to the residue field KA = Fp,t and if

(C

:

z)

= {e € , A n t : z e E C), then there e x i s t a tower of linear codes over KA

j ) , and the code C is said to be of type (k,, kl,.

.. ,

k a - l ) .

- -

I

? =

(e

^{: PO) g (C} : p i )

c .

^{f .}

E

(C : p - l }

such that

-

(C : p 3 ) has generator matrix The code C is said to be -T T

... xj 1 .

The ring A ⁼ GR@a,

fZ)

is a simple and separable exten- sion of B. = GR(p", f), which means there exists an invert- ible element

<

^{E A such that A = B(<)} and the residue field is a free-basis for A over B. We use g to denote the rep- .resentation of ^I E 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 K A = KB(T); with

K3

=

F,,.

Thus the set (1,

c, ... ,<'-'I

over B by the map C = [CO, Cl,.

...

C,,,-l] f A"* g oing . 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,}

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.

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) i s q d to the tmnsmit-diversity gain of the corresponding nt x 1 STBC ob- tained 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

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 vet- tor 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 p a n m e t e r s , the base ring GR(pa,f) i s the finite field F32 and the extension ring is the finite field F p . The maximal rate is then equal to R = nt

-

^{d -k 1 = 1. We}

use the primitive polynomial @ ( x ) = x4

+

^x

+

^{2 for con-}

structing 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.

In this ezample, the minimal polynomial of

5

^{O V ~ T}

Fg

is

-

@I(X) = x

-

Cwx

+

^{[lo (divides @e(x)).} The code

4

= {PIS) = (so i- s10x

I

^{so7 SI E F9) I}

and the corresponding m a t k representation i s

This

code is a n

MRD

code. Since p = 3 and f ⁼ 2, we can use the signal set RGR = ( m

+

^{in : -i}

5

^{m,n 5 1) in}

Ezample 3. The codewords of this

STBC

are of the form

100 division algebras,” IEEE %ns. Inform. Theory, vol.

49, no. 10, pp. 2596-2616, Oct 2003.

91 3 10’’

t

P 0

d

^IO“

[2] S.M. Alamouti, “A simple transmit diversity technique for wireless communication,” IEEE JSAC, vol. 16, no.

8, pp. 1451-1458, OCt 1998.

131 V. Tarokh,

N.

Sheshadri, and A.R. Calderbank, “Space- time codes for high data rate wireless communication:

., Performance criterion and code construction,” IEEE

5 10 15 za 25 Trans. Inform. Theory, vol. 44, pp. 744-765,

Mar

1998.

[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 con-

STBCs structions for PSK modulation,” IEEE Trans. Infom.

Theory, vol. 49, no. 10, pp. 2747-2751, Oct 2003.

. . . . . . 1 0-a

SNR (dB)

1 oa

[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.

1. pp. 3-16, .Jan-Mar 1985.

... ^; ... . . ; . ... ; ... ; ...

I

15 20 25 30 35

10-1 10

SNR (dB)

Figure 5: Performance comparison for nt = 1 = 2 and n, = 1 STBCs

where SO, S I E ( O , l , <lo,

c2’,

^{. . . . C70} and the map}

4

^{is tabu-}

iuted below.

In Figure

4 ,

we compare the performance of this code with the full-rank optimal STBCs obtained by the GU constmc- tzon for nt = I = 2 and n, = 1. We consider two diferent codes that ^QX constructible using

GU

construction: one cor- responding to the parameters p = 3, K = 2,U = 1 uses the

9-PSK

constellation and the other corresponding to the pa- m 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.

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 same rate and diversity for nt = 1 = d = 2 and n, = 1.

Clearly, our code with better coding gain outperforms the GU constructed cods.

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