Estimates for the growth of inverse determinant sums of quasi-orthogonal and number field lattices

0.11 MB·English

by  Roope Vehkalahti

Estimates for the growth of inverse determinant
sums of quasi-orthogonal and number field lattices
Roope Vehkalahti Laura Luzzi
Department of mathematics, University of Turku Laboratoire ETIS, CNRS - ENSEA - UCP
Finland Cergy-Pontoise, France
Email: roiive@utu.fi [email protected]
Abstract—Inverse determinant sums appear naturally as a that our upper bound is tight in the case of principal ideal
tool for analyzing performance of space-time codes in Rayleigh domains.
fading channels. This work will analyze the growth of inverse
determinant sums of a family of quasi-orthogonal codes and II. INVERSEDETERMINANTSUM
will show that the growths are in logarithmic class. This is
5 We begin by providing basic definitions concerning matrix
considerably lower than that of comparable numberfield codes.
1 lattices and spherical constellations, that are needed in the
0
sequel.
2
I. INTRODUCTION

n A. Matrix lattices and spherically shaped coding schemes
In [7] inverse determinant sums were proposed as a tool
a
J to analyze the performance of algebraic space-time codes Definition 2.1: A space-time lattice code C ⊆Mn(C) has
for MIMO fading channels. These sums can be seen as a the form
8
generalization of the theta series for the Gaussian channel. ZB1⊕ZB2⊕···⊕ZBk,

] They arise naturally from the union bound on the pairwise
T where the matrices B ,...,B are linearly independent over
error probability for spherical constellations, but also in the 1 k
I R, i.e., form a lattice basis, and k is called the rank or the
. analysis of fading wiretap channels [6].
s dimension of the lattice.
c In [7] the authorsanalyzed the growth of the inverse deter-
Definition 2.2: If the minimum determinant of the lattice
[ minantsumsofdiagonalnumberfield codesandof mostwell
L⊆M (C) is non-zero, i.e. it satisfies
known division algebra codes. In this work we are going to n
1
v extendthe analysisto a largeclass of quasi-orthogonalcodes. inf |det(X)|>0,
3 Our work will reveal that the growth of inverse determinant 06=X∈L
7 sumsoftheanalyzedcodesisconsiderablysmallerthanthatof wesay thatthecodehasanon-vanishingdeterminant(NVD).
7
thecorrespondingdiagonalnumberfieldcodes.Thisdifference Wenowconsideracodingschemebasedonak-dimensional
1
suggestthatasymptotically,withgrowingconstellation,quasi- lattice L inside M (C). For a given positive real number M
0 n
. orthogonal codes are considerably better than number field we define the finite code
1
codes. This difference can not be captured in the framework
0 L(M)={a|a∈L,kak ≤M},
5 of diversity-multiplexing gain tradeoff. F
1 For related work, we refer the reader to [1] and [4]. wherekak referstotheFrobeniusnorm.Inthefollowingwe
F
v: Remark 1.1: This is a corrected and extended version of will also use the notation
i [2]. The main correction is to Theorem 3.4 which we quoted
X B(M)={a|a∈M (C), kak ≤M},
carelessly in the previous version. This correction does affect n F
r all the formulasthat are derived from it. However,it doesnot
a for the sphere with radius M.
change any formulas related to complex constellations. In the
Let L ⊆ M (C) be a k-dimensional lattice. For any fixed
n
case of totally real fields the change sometimes doubles the m∈Z+ we define
number of units. The other correction is to the definition of
1
the normalized inverse determinant sum (1). This correction Sm(M):= .
L |det(X)|m
does not change any results, but is corrected for consistency. X∈LX(M)\{0}
The extensions in this paper are Propositions 3.8 and 3.6.
Our main goal is to study the growth of this sum as M
These results follow from improving on the estimate for
increases. Note, however, that in order to have a fair com-
the truncated Dedekind zeta function done in [7]. The new
parison between two different space-time codes, these should
estimate is Lemma3.3. The proofof this resultis due to Tom
be normalized to have the same average energy. Namely, the
Meurman and the authors are grateful to him.
volume Vol(L) of the fundamental parallelotope
Remark 1.2: We note that after the publication of [2] the
authors in [10] extended Proposition 3.5 to ideals and proved P(L)={α B +α B +...+α B |α ∈[0,1) ∀i}
1 1 2 2 k k i
should be normalized to 1. The normalized version of the the ring of algebraic integers in K. If B is an integral ideal
inverse determinant sums problem is then to consider the of K we will use the notation N(B)=[O :B].
K
growth of the sum S˜m(M) = Sm(M) over the lattice L˜ = Let I be the set of nonzero ideals of the ring O . The
L L˜ OK K
Vol(L)−1/kL. Since L˜(M) = Vol(L)−1/kL(MVol(L)1/k), Dedekind zeta function of the number field K is
we have 1
ζ (s)= , (2)
S˜Lm(M)=Vol(L)mn/kSLm(MVol(L)1/k). (1) K A∈XIOK N(A)s
Remark 2.1: In [2] we erroneously stated that where s is a complex number with ℜ(s)>1. For a truncated
version of this sum we will use notation
S˜m(M)=Vol(L)mn/kSm(M).
L L 1
ζ (s,M):= .
Thecorrectformis(1),whichwasoriginallystatedin[9].The K N(A)s
X
difference is that in the correct expression the radius of the A∈IOK,N(A)≤M
ball also increases. However, this difference does not affect In the following we will need an estimate for ζK(s,M). We
theresultsinthispaper.Ineverycase inthispaperthegrowth will begin with a classical result. Let us suppose we have an
of the inverse determinant is logarithmic and therefore the algebraic number field K with signature (r1,r2).
constanttermVol(L)1/k willhavenoeffectonthedominating Theorem 3.1: The function ζK(s) converges for all s,
term of the sum. ℜ(s)>1 and has a simple pole at 1 with residue
2r1(2π)r2h R
B. Inversedeterminantsumsanderrorperformanceofspace- lim(s−1)ζ (s)= K K,
K
time lattice codes s→1 ω |d(K)|
p
Let us now consider the slow Rayleigh fading MIMO where RK is the regulator of the number field K, d(K) the
channelwith ntransmitandn receiveantennas.Thechannel discriminant and ω the number of roots of unity in K.
r
equation can then be written as In the following we will use the shorthand
2r1(2π)r2R
Y = HX+N, α = K.
K
ω |d(K)|
where H and N are respectively the channel and noise
InordertoestimatethetrunpcatedDedekindzetafunctionat
matrices.WesupposethatthetransmittedcodewordX belongs
the point1 we needan estimate forthe numberof ideals with
toafinitecodeL(M)⊂M (C)carvedfromak-dimensional
n bounded norm. Let us denote with N(K,M) the number of
NVDlatticeLasdefinedpreviously.Intermsofpairwiseerror
integral ideals of K with norm less than or equal to M. We
probability, we have for X 6=X′
then have the following.
P(X →X′)≤ 1 , Proposition 3.2: [5, Theorem 5] Given a degree n number
|det(X −X′)|2nr field K there exist positive constants c and a, independentof
andthecorrespondingupperboundonoverallerrorprobability M, such that
1 |N(K,M)−αKhKM|≤cM1−a.
P ≤ .
e X |det(X)|2nr We are now ready to estimate the truncated Dedekind zeta
X∈L,0<kXkF≤2M
function.
Our main goal is now to study the growth of the sum
Lemma 3.3: We have the following
Sm(M) as M increases. In particular, we want to find, if
L 1
possible, a function f(M) such that =h α logM +O(1).
K K
N(A)
X
Sm(M)∼f(M). A∈IOK,N(A)≤M
L
Proof: Let us first write
III. INVERSEDETERMINANTSUMS OF ALGEBRAIC
1 1 z(n)
NUMBERFIELD CODES = 1= .
N(A) n n
Inthissectionwewillgiveandreviewsomeresultsconcern- N(XA)≤M nX≤M N(XA)=n nX≤M
ing inverse determinantsums of diagonalnumber field codes.
Manipulating the previous expression, we can write it in the
These results will play an important role in our analysis of
form
quasi-orthogonal codes. The proofs are mostly analogous to
z(n)−h α h α
those given in [7] and we will skip them. However, the case S+T = K K + K K. (3)
n n
where the receiver has a single antenna will improve on the nX≤M nX≤M
original one given in [7]. Unlike in the rest of the paper we LetusnowanalyzethefirsttermS byusingpartialsummation
willstatetheresultsinthenormalizedfromS˜Lm(M)following [12, A. 21]
the general normalization given in [9].
1 M 1
We need first some preliminary results. Let us suppose we S = (z(n)−h α )+ (z(n)−h α )dt.
have an algebraic number field K. We will denote with OK M nX≤M K K Z1 t2 Xn≤t K K
According to Proposition 3.2 we have that where c = hK2nnn.
K (n−1)!
Proof:FollowingtheproofofProposition4.4andSection
(z(n)−h α )= z(n)−h α t+O(1)=O(t1−a).
K K K K 4.C in [7] we have an estimate
Xn≤t Xn≤t
Mn
It follows that S˜ψm(OK)(M)≤ζK(1,nn/2)(N˜K(logM)n−1+O((logM)n−2)),
M 1 M and the end result now follows from plugging the result of
(z(n)−h α )dt= O(t−1−a)dt.
Z t2 K K Z Lemma 3.3 and Theorem 3.4 to this estimate.
1 Xn≤t 1
For principalideal domains the upper bound is likely tight.
Putting now all together we have that
B. Inversedeterminantsumsofcomplexdiagonalnumberfield
1 M codes
S = O(M1−a)+ O(t−1−a)dt=O(M−a)+O(1).
M Z1 Let K/Q be a totally complex extension of degree 2n and
Estimating hKαK = h α logM +O(1), gives us {σ1,...,σn} be a set of Q-embeddings, such that we have
n≤M n K K chosen one from each complex conjugate pair. Then we can
the final resPult.
Remark 3.1: This results improves on the very crude esti- define a relative canonical embedding of K into Mn(C) by
mate [7]
ψ(x)=diag(σ (x),...,σ (x)).
1 n
1
≤d(logM)n, The ring of algebraic integers O has a Z-basis W =
K
N(A)
A∈IOKX,N(A)≤M {w1,...,w2n} and ψ(W) is a Z-basis for the full lattice
ψ(O ) in M (C).
where d is some constant. K n
Proposition 3.7: [7, Section 4.C] Let K be a totally com-
A. Inverse determinant sums of real diagonal number field plex algebraic number field of degree 2n. If nr >1, we have
codes that
Let K be a totally real number field of degree n and let S˜2nr (M)≤N˜ ζ (n )(logM)n−1+O((logM)n−2)
ψ(OK) K K r
{σ ,··· ,σ } be the Q-embeddings from K to R. We then
1 n
and
have the canonical embedding ψ :K 7→M (R) defined by
n
N˜ (logM)n−1+O((logM)n−2)≤S˜2nr (M),
ψ(x)=diag(σ1(x),...,σn(x)). K ψ(OK)
It is a well known result that ψ(OK) is an n-dimensional where N˜K = ωR((nn)−n1−)1!(2−n |d(K)|)nr.
NVDlatticeinM (R).Letusnowconsiderthecorresponding Proposition 3.8: Let Kpbe a totally complex algebraic
n
inverse determinant sum. The main role in the analysis is number field of degree 2n. We then have
played by the following unit group density result. S˜2 (M)≤c (logM)n+O((logM)n−1) (4)
Theorem 3.4 ([8]): Let us suppose that [K : Q] = n, we ψ(OK) K
then have that and
|ψ(O∗ )∩B(M)|=N (logM)n−1+O((logM)n−2), N˜ (logM)n−1+O((logM)n−2)≤S˜2 (M), (5)
K K K ψ(OK)
where NK = Rω(nnn−−11)!. where cK = hK(nπ−n12)n!n and N˜K = ωR((nn)−n1−)1!(2−n |d(K)|).
Here R is the regulator of the number field K, ω the number Proof: Following the proof of Propositionp4.4 in [7] we
of roots of unity in K. have an estimate
We first have a real analogue to the results in [7].
M2n
Proposition 3.5: Let us suppose that K is a totally real S˜ψm(OK)(M)≤ζK(1, nn )(N˜K(logM)n−1+O((logM)n−2),
number field with [K :Q]=n and that m>1. Then
and the end result now follows from plugging the result of
S˜m (M)≤N˜ ζ (m)(logM)n−1+O((logM)n−2) Lemma 3.3 and Theorem 3.4 to this estimate
ψ(OK) K K
Again for principal ideal domains the upper bound is likely
and
tight.
N˜ (logM)n−1+O((logM)n−2)≤S˜m (M),
K ψ(OK) IV. QUASI-ORTHOGONAL CODES FROM DIVISION
where N˜ = ω(n)n−1 ( |d(K)|)m. ALGEBRAS
K RK(n−1)! In the following we are considering the Alamouti-like
In the case of singlepreceive antenna we now have the
multiblock codes from [3]. With respect to their complexity
following.
and other properties, all of the codes of this type are quasi-
Proposition 3.6: Let us suppose that K is a totally real
orthogonal.It is even possible to provethat many of the fully
number field with [K :Q]=n and that m=1. Then
diverse quasi-orthogonal codes in the literature are unitarily
S˜m (M)≤c (logM)n+O((logM)n−1), equivalent to these multi-block codes. In the following we
ψ(OK) K
willuseseveralresultsandconceptsfromthetheoryofcentral where ℜ(s)>1 and I is the set of right ideals of Λ. When
Λ
simplealgebras.Wereferthereaderto[11]foranintroduction ℜ(s)>1, this series is converging.
to this theory. TheunitgroupΛ∗ ofanorderΛconsistsofelementsx∈Λ
Let us consider the field E = KF that is a compositum such that there exists a y ∈Λ with xy =1 . Another way to
A
of a complex quadratic field F and a totally real Galois define this set is Λ∗ ={x∈Λ||detψ(x)|=1}.
extension K/Q of degree k. We suppose that K ∩F = Q,
A. Inverse determinant sums of quasi-orthogonalcodes
Gal(F/Q)=<σ > and Gal(K/Q)={τ ,τ ,...,τ }. Here
1 2 k
Let us suppose that K, D and Λ are as in the previous
σ is simply the complex conjugation. We can then write that
Gal(FK/Q)=Gal(K/Q)⊗<σ >. section and that [K : Q] = k. We then have that φ(Λ) is a
4k-dimensional NVD lattice in M (C) and we can consider
Let us now consider a cyclic division algebra 2k
the growth of the sum
D=(E/K,σ,γ)=E⊕uE,
1
=S2nr (M).
where u∈D is an auxiliarygeneratingelementsubjectto the X |detψ(x)|2nr ψ(Λ)
ψ(x)∈ψ(Λ)(M)
relations xu = ux∗ for all x ∈ E and u2 = γ ∈ O , where
K Just as in [7] the previoussum can be analyzedfurtherinto
()∗ is the complex conjugation.We can consider D as a right
vector space over E and every element a = x1 +ux2 ∈ D S2nr (M)= |ψ(xΛ∗)∩B(M)|, (8)
maps to ψ(Λ) |det(ψ(x))|2nr
X
x∈X(M)
x x
φ(a)= 1 2 .
(cid:18)γx∗ x∗(cid:19) where X(M) is some collection of elements x∈Λ such that
2 1
kψ(x)k ≤M, each generating a different right ideal.
This mapping can then be extended into a multi-block F
representation φ:D 7→M (C). B. Uniform upper and lower bounds for |ψ(xΛ∗)∩B(M)|
2k
The keyelementin the analysisof|ψ(xΛ∗)∩B(M)| is the
ψ(a)=diag(τ (φ(a)),τ (φ(a))...,τ (φ(a))). (6)
1 2 k following.
Lemma 4.3 (Eichler): The unit group Λ∗ has a subgroup
Example 4.1: Inthe case wherek =2 eachelementa∈D
gets mapped as O∗ ={x|x∈Λ∗,x∈O },
K K
x1 x2 0 0 and we have [Λ∗ :O∗ ]<∞.
ψ(a)=γx∗2 x∗1 0 0 . Letj =[Λ∗ :OK].KBychoosingaset{a1,...,aj}ofcoset
0 0 τ(x1) τ(x2) leaders of O∗ in Λ∗, we have that
  K
 0 0 τ(γx2)∗ τ(x1)∗
  j
Inordertobuildaspace-timelatticecodefromthedivision |ψ(xΛ∗)∩B(M)|≤ |ψ(xa O∗ )∩B(M)|. (9)
i K
algebra D we will need the following definition. Xi=1
Definition 4.1: Let OK be the ring of integers of K. An In order to give an uniform upper bound for |ψ(xΛ∗) ∩
OK-order Λ in D is a subringof D, havingthe same identity B(M)|, it is now enough to give a uniform upper bound for
element as D, and such that Λ is a finitely generated module |ψ(xa O∗ )∩B(M)|.Beforestatingourmainresultsweneed
i K
over OK and generates D as a linear space over K. few lemmas. We will skip the proofs of some of them.
Let us suppose that Λ is an OK-order in D. We call φ(Λ) Lemma 4.4: Let us suppose that A is a diagonal matrix in
an order code. In the rest of this paper, we suppose that the M (C) with |detA|≥1. We then have that
n
divisionalgebrasunderconsiderationareoftheprevioustype.
|Aψ(O∗ )∩B(M)|≤|ψ(O∗ )∩B(cM)|,
Lemma 4.1: If Λ is an O -order in D K K
K
where c is a fixed constant, independent of A and M.
|det(φ(x))| = [Λ:xΛ], (7)
Lemma 4.5: Let us suppose that x and y are elements in
p
where x is a non-zero element of Λ. OKF, we then have that
Lemma 4.2: Let us suppose that Λ is a OK-order of a |ψ(x)ψ(O∗ )∩B(M)|≤|ψ(O∗ )∩B(cM)|,
K K
division algebra D with center K of degree k and that φ is
a multi-block representation. Then the order code φ(Λ) is a and
4k-dimensional lattice in the space M2k(C) and |ψ(uy)ψ(O∗ )∩B(M)|≤|ψ(O∗ )∩B(cM)|,
K K
detmin(ψ(Λ))=1. where c is a real constant independentof x,y and M.
Proof: The first result is simply Lemma 4.4 and the
Let D be an index-n K-central division algebra and Λ a
second follows as ψ(u) is a fixed matrix.
O -order in D. The (right) Hey zeta function of the order Λ
K Lemma 4.6: Let us suppose that x and y are elements in
is
1 E. We then have that
ζ (s)= ,
Λ IX∈IΛ [Λ:I]s ||ψ(x)+ψ(uy)||2F =||ψ(x)||2F +||ψ(uy)||2F.
Proof: By an elementary calculation we see that < where n >1 and R and ω are the regulator and the number
r
ψ(x),ψ(uy)>=0 and the claim follows. of roots of unity in the center K.
Proposition 4.7: Let us suppose that x ∈ Λ, we then have Proof: As previously mentioned, we can imitate [7] to
that get
|ψ(xΛ∗)∩B(M)|
|ψ(x)ψ(OK∗ )∩B(M)|≤|ψ(OK∗ )∩B(cM)|, Sψ2n(Λr)(M)= |det(ψ(x))|2nr . (10)
X
x∈X(M)
where c is a constant independent of M and x.
According to Lemma 4.1 we have that |det(ψ(x))|2nr =[Λ:
Proof: Let us suppose first that x = x +ux , where
1 2
x ∈O and where u2 ∈O . According to Lemma 4.6, we xΛ]nr. Now
i E K
have that 1 1
≤ ≤ζ (n ).
||ψ(x)ψ(y)||2 =||ψ(x1)ψ(y)||2+||ψ(ux2)ψ(y)||2, x∈XX(M)|det(ψ(x))|2nr x∈XX(M)[Λ:xΛ]nr Λ r
ApplyingthisinequalitywithProposition4.8to(10)nowgives
for any y ∈O .
E
us the final result.
Therefore if ψ(x)ψ(y)∈B(M), then also
V. QUASI-ORTHOGONAL CODES ARE BETTERTHAN
ψ(x )ψ(y)∈ B(M)andψ(ux )ψ(y) ∈B(M).
1 2
DIAGONAL NUMBER FIELDCODES
It follows that we can upper bound |ψ(x)ψ(OK∗ )∩B(M)| Let us now suppose we have an n×nr-MIMO channel,
with (forsimplicityweassumen >1).Fortheexistenceofquasi-
r
orthogonalcodewealsohavetoassumethat2|n.Letusnow
max{|ψ(x )ψ(O∗ )∩B(M)|,|ψ(ux )ψ(O∗ )∩B(M)|}.
1 K 2 K compare the growth of determinant sums of quasi-orthogonal
According to lemma 4.5 we then have that and comparable diagonal number field codes in this n×n -
r
MIMO channel.
|ψ(x)ψ(O∗ )∩B(M)|≤|ψ(O∗ )∩B(cM)|.
K K In order to build a quasi-orthogonal code ψ(Λ) in M (C)
n
Let us now suppose that Λ is a general order in D. As ψ(Λ) the center K of the algebra D must be an n/2-dimensional
is finitely generated as an additive group in M (E), we can totally real number field. For a number field code ψ(OL) ⊆
n
choose an integer d such that dψ(Λ) ⊆ ψ(O )+ψ(uO ). Mn(C), the field L must be an n-dimensional extension of
E E
The result now follows from the previous consideration. some complex quadratic field F.
As we earlier saw, we have that
Proposition 4.8: Using the previous notation we have
1
|ψ(xΛ∗)∩B(M)|≤[Λ∗ :O ]· =θ(|ψ(Λ∗)∩B(M)|)
K |det(X)|2nr
ω(k)k−1 X∈ψX(Λ)(M)
logMk−1 +O(logMk−2).
R(k−1)! and
Proof:Letj =[Λ∗ :O ].Bychoosingaset{a ,...,a } |ψ(Λ∗)∩B(M)|=θ(|ψ(O∗ )∩B(M)|)=θ(logMn/2−1).
K 1 j K
of coset leaders of O∗ in Λ∗, we then have that
K Therefore
j 1
|ψ(xΛ∗)∩B(M)|≤ |ψ(xaiOK∗ )∩B(M)|. |det(X)|2nr =θ(logMn/2−1).
X
Xi=1 X∈ψ(Λ)(M)
According to Proposition 4.7 we then have that On the other hand for the number field code we have that
1
|ψ(xΛ∗)∩B(M)|≤[Λ∗ :OK]||ψ(OK∗ )∩B(cM)|. |det(X)|2nr =θ(|ψ(OL∗)∩B(M)|)
X
X∈ψ(OL)(M)
ApplyingTheorem3.4tothisequation,wegetthefinalresult.
=θ(logMn−1).
Here the last result follows from [8, Theorem 2].
C. Upper and lower bounds for inverse determinant sums of
We can now see that the growth of the inverse determinant
quasi-orthogonalcodes
sum for the quasi-orthogonalcode is considerably lower than
Proposition 4.9: Let us suppose that [K : Q] = k and set that of the number field code. This is due to the fact that the
n = 2k. We then have that ψ(Λ) is a 2n-dimensional lattice unitgroupoftheorderΛisessentiallythatofalowdegreereal
in M (C) and numberfield. We notethatthis differencecan notbecaptured
n
in the context of DMT as both of these codes have the same
ω(n)n/2−1
logMn/2−1 2 +O(logMn/2−2)≤S2nr (M) DMT curve.
R(n/2−1)! ψ(Λ)
ACKNOWLEDGEMENT
≤ζΛ(nr)[Λ∗ :OK]logMn−22ωR((n2n)−n2−2)2! +O(logMn/2−2), ofTFhinelarensdeagrrcahntof#2R5.2V4e5h7k.alahtiissupportedbytheAcademy
2
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