Number-Theoretic Fast-Decodable Space–Time … · Number-Theoretic Fast-Decodable ... {Time Codes from Cyclic Division Algebras ... Number-Theoretic Fast-Decodable Space–Time

headlinecolor

Number-Theoretic Fast-DecodableSpace–Time Codes for Multiuser

Communications

Camilla Hollantiwith Amaro Barreal and Nadya Markin

Credit for slides: Amaro

Aalto UniversityDepartment of Mathematics and Systems Analysis

Finland

York 2016

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Outline headlinecolor

Overview

1 Space–Time CodingSpace–Time Codes from Cyclic Division AlgebrasOn Fast-DecodabilityIterative Code Construction

2 Amplify-and-Forward Relaying

3 Explicit ConstructionsCode Construction for SIMO-NAFCode Construction for MIMO-NAF

4 Further Applications and Conclusions

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1. Space–Time Coding headlinecolor





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Need for diversity

Waves transmitted over a wireless channel suffer from environmentaleffects and fading. Combat those effects through diversity!

I Spatial diversity: Multiple antennas at the transmitter and/orreceiver.

I Temporal diversity: Use different time slots for transmission.

⇒ space–time codes.

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MIMO channel model

Consider an ns × nd-antenna system:

⇔ H =

h11 h12 · · · h1ns

h21 h22 · · · h2ns

......

. . ....

hnd1 hnd2 · · · hndns

Transmission using T time slots can be modeled as

Y = HX +N.

I The channel matrix H ∈ Mat(nd×ns,C) models Rayleigh fading.I X ∈ Mat(ns × T,C) is a codeword matrix.I N ∈ Mat(nd × T,C) a noise matrix, whose entries are complex

Gaussian with zero mean.

We will focus on square codes, ns = T .5 / 40

1. Space–Time Coding headlinecolorSpace–Time Codes from Cyclic Division Algebras





F. Oggier, J-C. Belfiore, and E. Viterbo. “Cyclic division algebras: a tool for space-time coding”. In: Foundations and trends in communications and informationtheory 4.1 (2007), pp. 1–95

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Algebraic setup

I L/K degree n cyclic Galois extension of number fields withrespective ring of integers OK and OL.

I Cyclic Galois group Γ (L/K) = 〈σ〉.I Choose an OK-basis {ω0, . . . , ωn−1} of OL (assume K is a PID).

I Fix γ ∈ K× such that γi /∈ NmL/K(L×), i = 1, . . . , n− 1.

The triple

C = (L/K, σ, γ) ∼=n−1⊕i=0

uiL

is a cyclic division algebra (CDA) of degree n, andmultiplication is determined by the relations

un = γ, lu = uσ(l) for all l ∈ L.

C

L

n

K

n

Qm

C

OL

{ui

}n−1

i=0

OK

{ωi}n−1i=0

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The left-regular representation

We need to identify elements x =∑nj=0 u

jxj ∈ C with matrices.

The (transposed) left-regular representation (LRR) λ : C → Mat(n,C)is an injective algebra homomorphism, given by

λ(x) =

x0 x1 · · · xn−2 xn−1

γσ(xn−1) σ(x0) · · · σ(xn−3) σ(xn−2)...

......

...γσn−2(x2) γσn−2(x3) · · · σn−2(x0) σn−2(x1)γσn−1(x1) γσn−1(x2) · · · γσn−1(xn−1) σn−1(x0)

.

Write xj =∑n−1i=0 ωixji ∈ OL, where xji ∈ OK . If we further expand

using a Z-basis, we see that λ(x) carries mn2 independent integers(e.g., PAM symbols).

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Space–time codes and their properties

Definition

A space–time (ST) code constructed from a CDA C is a subset

X ⊂finite

Im(λ(C)) = {λ(x)|x ∈ C} .

Often, a shaping element α ∈ L× is introduced to restrict the entriesto an ideal (α) ⊂ OL and improve the performance of the code, whichcan have many desirable properties, such as

I Full-rank codeword matrices(cf. diversity gain),

I Non-vanishing determinants(cf. coding gain),

I Balanced energy amongtransmitters (cf. PAPR),

I Optimal DMT,

I Fast-decodability, etc.

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Equivalent representation

We can give a more explicit presentation of a ST code in terms of itsgenerating matrices as follows.

Definition

Let {Bi}ki=1 be a set of fixed ns × T complex weight matrices. Alinear space–time block code of rank k is a set of the form

X =

{k∑i=1

siBi

∣∣∣∣∣si ∈ S ∩ Z

},

where S ⊂ Z is the finite signaling alphabet in use.

Remark

This definition agrees with the previous one by setting ns = T = n andchoosing the set of weight matrices {Bi} to be a basis of C over Q.

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Full-diversity codes with NVD

Motivation for the use of division algebras in space–time coding:

Lemma

Let D be a division algebra and K a field. If φ : D → Mat(n,K) is aring homomorphism and X ⊆ φ(D) is any finite subset, thenrank(X −X ′) = n for any distinct X,X ′ ∈ X .

λ is a ring homomorphism, and we defined X ⊂ Im(λ(C)) to be finite.

Lemma

Let X be a space–time code coming from an order O of a CDAC = (L/K, σ, γ) with center K = Q or K = Q(

√−m), m ∈ Z>1

squarefree. Then X has the non-vanishing determinant property.

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Example: the Golden code

Fix n = 2. The algebraic setup for the Golden code is as follows:

I L/K = Q(i,√

5)/Q(i).

I Ring of integers OK = Z[i], OL = OK [ω], where ω = 1+√

52 .

I Cyclic Galois group Γ (L/K) = 〈σ :√

5 7→ −√

5〉.I Our CDA is the triple CG = (L/K, σ, i) ∼= L⊕ uL, with u2 = i.

I Choose the shaping element α = 1 + i− iω.

Restricting the entries to OL, the Golden code is a finite subset

XG ⊂{

1√5

[α

σ(α)

] [x0 x1

iσ(x1) σ(x0)

]∣∣∣∣xi ∈ OL}=

{1√5

[α

σ(α)

] [x00 + ωx01 x10 + ωx11

i(x10 + σ(ω)x11) x00 + σ(ω)x01

]∣∣∣∣xij ∈ OK} .

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A slight modification

We will mostly consider field extensions L/K of degree 2. To havebalanced energy and good decodability, we slightly modify the matrixrepresentation of the elements in C = (L/K, σ, γ).

Instead of representing x = c+√γd ∈ O ⊂ C using the representation

λ(x) over the maximal subfield K, we define

λ̃ : x 7→[

c −√−γσ(d)√−γd σ(c)

].

This function is commonly used, and maintains the originaldeterminant.

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1. Space–Time Coding headlinecolor On Fast-Decodability





G. Berhuy, N. Markin, and B. A. Sethuraman. “Bounds on Fast Decodability ofSpace-Time Block Codes, Skew-Hermitian Matrices, and Azumaya Algebras”. In:IEEE Trans. Inf. Theory 61.4 (2015), pp. 1959–1970E. Biglieri, Y. Hong, and E. Viterbo. “On fast-decodable space-time block codes”.In: IEEE Trans. Inf. Theory 55.2 (2009), pp. 524–530

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Complexity of ML-decoding

Given a space–time code X , Maximum-Likelihood (ML) decodingamounts to finding the codeword X ∈ X that minimizes

δ(X) := ||Y −HX||2F .

By defining the real-valued matrix B :=[vec(HB1) . . . vec(HBk)

],

we can reduce the decoding problem to read

arg minX∈X

{||Y −HX||2F

}; arg min

s∈Sk

{|| vec(Y )−Bs||2E

}(B = QR) ; arg min

s∈Sk

{||Q† vec(Y )−Rs||2E

}.

Definition

The ML decoding complexity of a rank-k ST code X is upper boundedby the worst-case (ML) complexity |S|k corresponding to anexhaustive search. A ST code X is said to be fast-decodable if itsworst-case ML decoding complexity is of the form |S|k′ for k′ < k − 1.

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Hurwitz-Radon quadratic form

Introducing the R-matrix in the decoding process permits to directlyread out the decoding complexity of a given code.

Definition

The Hurwitz-Radon Quadratic Form (HRQF) is the map

Q : X → R; X 7→∑

1≤i≤j≤ksisjmij ,

where mij := ||BiB†j +BjB†i ||2F and si ∈ S.

I Define the matrix M = (mij). Then mij = 0 if and only if

BiB†j +BjB

†i = 0, that is, if Bi and Bj are mutually orthogonal.

I Premultiplication of the weight matrices by H does not affect thezero structure of M , whereas it does affect that of R.

I Yet, the zero structure of the R and M matrices are convenientlyrelated to each other.

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Hurwitz-Radon quadratic form

Introducing the R-matrix in the decoding process permits to directlyread out the decoding complexity of a given code.

Definition

The Hurwitz-Radon Quadratic Form (HRQF) is the map

Q : X → R; X 7→∑

1≤i≤j≤ksisjmij ,

where mij := ||BiB†j +BjB†i ||2F and si ∈ S.

I Define the matrix M = (mij). Then mij = 0 if and only if

BiB†j +BjB

†i = 0, that is, if Bi and Bj are mutually orthogonal.

I Premultiplication of the weight matrices by H does not affect thezero structure of M , whereas it does affect that of R.

I Yet, the zero structure of the R and M matrices are convenientlyrelated to each other.

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Breaking News!

Recently, the criteria for fast-decodability have been revisited:

A. Mejri, M. Kshiba, and G. Rekaya. “Reduced-Complexity MLDecodable STBCs: Revisited Design Criteria”. In: IEEEISWCS. 2015, pp. 666–670.

The authors show that the HRQF method does not capture allpossible fast-decodable code families, and propose a relaxedcriterion on the mutual orthogonality condition.


(Conditional) g-group decodability

Definition

A rank-k ST code X is conditionally g-group decodable if there existsa partition of {1, . . . , k} into g + 1 ≥ 3 disjoint subsets Γ1, . . . ,Γg, ΓX ,

such that BiB†j +BjB

†i = 0 for i ∈ Γp, j ∈ Γq, 1 ≤ p < q ≤ g.

X is g-group decodable if |ΓX | = 0.

The R-matrices related to these families of codes are of the form

Rcond =

D1 G1

. . ....

Dg GgG

resp. R =

D1

. . .

Dg

,where Di ∈ Mat(|Γi|,R), G ∈ Mat(|ΓX |,R) upper triangular.

Lemma

The ML-decoding complexity of X is |S||ΓX |+ max

1≤i≤g|Γi|

.

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1. Space–Time Coding headlinecolorIterative Code Construction





N. Markin and F. Oggier. “Iterated space-time code constructions from cyclicalgebras”. In: IEEE Trans. Inf. Theory 59.9 (2013), pp. 5966–5979

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1. Space–Time Coding headlinecolorIterative Code Construction

Iterative codes from CDAs

Let C = (L/K, σ, γ) be a CDA or degree n. Fix θ = ζθ′ ∈ C,τ ∈ AutQ(K). For X = λ(x), Y = λ(y) (LRR), define the followingfunction:

ατ,θ : C × C → Mat(2n,K)

(x, y) 7→[X ζ

√θ′τ(Y )√

θ′Y τ(X)

].

If the matrices {Bi}ki=1 define a ST code X , the iterated ST code Xit

is double the rank and defined by the matrices

{ατ,θ(Bi, 0), ατ,θ(0, Bi)} .

The importance of this construction is that by making certainassumptions, we can guarantee that the code Xit will inherit desirableproperties from X , such as fast-decodability.

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2. Amplify-and-Forward Relaying headlinecolor





S. Yang and J-C. Belfiore. “Optimal space-time codes for the MIMO amplify-and-forward cooperative channel”. In: IEEE Trans. Inf. Theory 53.2 (2007), pp. 647–663

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N-relay MIMO NAF channel

Relay 1

.

.

.

Relay N

Dest.

G1

GN

SourceF

H1

HN ns antennas at the source.nr ≤ ns antennas at each relay.nd antennas at the destination.

We impose the half-duplex constraint, i.e., a relay cannot receive andtransmit at the same time.

From the destination’s point of view this can be viewed as a virtualsingle-user MIMO channel

Ynd×n = Hnd×nXn×n + Vnd×n,

where n = 2Nns and H has a special structure.

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Frame model

I N consecutive cooperation frames define a superframe.

I The relays take turns to cooperate with the transmitter in theirrespective cooperation frame.

I All channels remain static for the entire superframe.

Source

R1

R2

...

RN

Dest.

X1,1 X1,2 X2,1 X2,2 · · · XN,1 XN,2

X1,1 X1,1

X2,1 X2,1

. . .

XN,1 XN,1

Y1,1

0

Y1,2

T2

Y2,1

T

Y2,2

3T2

2T

· · · YN,1 YN,2

NT

Transmitted and received signals are represented by solid and dashedboxes, respectively.

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Codeword structure

The overall (equivalent) codewords are of the form

X = diag {Ξi}Ni=1 =

Ξ1

. . .

ΞN

,where Ξi ∈ Mat(2ns,C).

It would be desirable to construct block-diagonal codes which have

I “full” rate 2nd (real) symbols per channel use (rspcu), i.e., thenumber of independent real information symbols (e.g., PAM) percodeword equals 4ndnsN ,

I full rank 2nsN ,

I non-vanishing determinants,

I fast(er) decoding.

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3. Explicit Constructions headlinecolor





A. Barreal, C. Hollanti, and N. Markin. “Fast-Decodable Space–Time Codes forthe N -Relay and Multiple-Access MIMO Channel”. In: IEEE Trans. WirelessCommun. 15.3 (2016), pp. 1754–1767

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3. Explicit Constructions headlinecolor

The map that does the trick

Definition

Consider an N -relay NAF channel. Given a ST code X ⊂ Mat(2ns,C)and a suitable function η of order N (i.e., ηN (X) = X), define thefunction

Ψη,N : X → Mat(nN,C)

X 7→ diag{ηi(X)

}N−1

i=0=

X 0 · · · 00 η(X) · · · 0...

. . ....

0 · · · ηN−1(X)

.

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3. Explicit Constructions headlinecolorCode Construction for SIMO-NAF





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Algebraic framework – SIMO

Assume ns = nr = 1, and nd ≥ 2, and consider the tower

C = (L/K, σ :√a 7→ −√

a, γ)

L = K(√a)

2

K = F (ξ)

2

F = Q(√−m)

N

Q2

Q(√a)

2N

2

We assume m ∈ Z≥1 and a ∈ Z\ {0}, both square-free.

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Infinite family for SIMO

Theorem

Consider the algebraic setup from above with a < 0, γ < 0. Fix agenerator 〈η〉 = Γ (K/F ) and define the set

X = {Ψη,N (X)}X∈λ̃(O) ={

diag{ηi(X)

}N−1

i=0

∣∣∣X ∈ λ̃(O)}.

The code X is of rank 8N , rate R = 4 rspcu and has the NVDproperty. It is full-rate if nd = 2. Moreover, X is conditionally4-group decodable, and its decoding complexity (up to a constant) canbe reduced from |S|8N to |S|5N , where S is the real constellation used,resulting in a complexity order reduction of 37.5%.

Moreover, full-rate codes constructed using this method will achievethe optimal DMT of the channel.

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A two-relay example code

Consider N = 2 relays and the following tower of extensions.

C = (L/K, σ :√−3 7→ −

√−3,− 2√

5)

L = K(√−3)

2

K = Q(i, ξ)

2

F = Q(i)

2

Q2

Q(√−3)

4

2

I For ξ =√

5, the algebra C isdivision.

I Let 〈η〉 = Γ (K/F ).

I X ∈ λ̃(OL) is of the form

X =

[c −√−γσ(d)√−γd σ(c)

]

Define the 2-relay code

X ={

Ψη,2(X)|X ∈ λ̃(OL)}

=

{[X

η(X)

]∣∣∣∣X ∈ λ̃(OL)

}.

This is a fully diverse NVD code of rank 16. It is conditionally 4-groupdecodable, and its decoding compexity is |S|10 in contrast to |S|16.

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What about performance?

We compare the performance of the example code with the optimalcode proposed in [Yang]) – a lifted version of the Golden code – andfurther an unshaped version of the same. (ns = nr = 1, nd = 4)

SNR0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

BLE

R

10-5

10-4

10-3

10-2

10-1

100

Non-FD, ScaledNon-FD, UnscaledFD, Unscaled

4-QAM

64-QAM

S. Yang and J-C. Belfiore. “Optimal space-time codes for the MIMO amplify-and-forward cooperative channel”. In: IEEE Trans. Inf. Theory 53.2 (2007), pp. 647–663 30 / 40

3. Explicit Constructions headlinecolorCode Construction for MIMO-NAF





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Algebraic framework – MIMO

Assume ns = 2, nd ≥ 1 and N = (p− 1)/2 relays (p ≥ 5 prime)equipped with nr ≤ 2 antennas.

C = (L/K, σ :√a 7→ −√

a, γ)

L = K(√a)

2

K = Q(ξ)

2

Q

NF = Q(√a)

N

2

I K = Q(ξ) = Q+(ζp) ⊂ Q(ζp)maximal real subfield of thepth cyclotomic field.(ξ = ζp + ζ−1

p )

I a ∈ Z\ {0} is square-free.

I 〈σ〉 = Γ(L/K) and〈η〉 = Γ(L/F ).

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Infinite family for MIMO

Theorem

Consider the algebraic setup from above, and choose a ∈ Z<0 suchthat p = aOK is a prime ideal. Fix further γ < 0 andθ ∈ OK ∩ R× = Z[ξ] ∩ R× such that

I γ and θ are both nonsquare mod p,

I the quadratic form 〈γ,−θ〉L is anisotropic,

and further let τ = σ. For O ⊂ C an order, the distributed ST code

X ={

Ψη,N (ατ,θ(X,Y )) = diag{ηi(ατ,θ(X,Y ))

}N−1

i=0

∣∣∣X,Y ∈ λ̃(O)}

is a full-diversity ST code of rank 8N , rate R = 2 rspcu, exhibits theNVD property and is FD. Its decoding complexity is |S|k′ , where

k′ =

{4N if a ≡ 1 mod 4,

2N if a 6≡ 1 mod 4,

resulting in a reduction in complexity of 50% and 75%, respectively.33 / 40


Two 3-relay example codes

Consider N = 3 relays and the following towers of extensions.

C1 = (L1/K, σ1 :√−3 7→ −

√−3,−1)

L1 = Q(√−3, ξ)

2

K = Q(ξ)

2

Q

3F1 = Q(√−3)

3

2

C2 = (L2/K, σ2 :√−5 7→ −

√−5,− 2

1+ξ )

L2 = Q(√−5, ξ)

2

K = Q(ξ)

2

Q

3F2 = Q(√−5)

3

2

I Both algebras are division for ξ = ζ7 + ζ−17 .

I For i = 1, 2, set τi = σi and 〈ηi : ξ 7→ ξ2 − 2〉 = Γ (Li/Fi).(η1 6= η2, since they have distinct fixed fields.)

I Let Oi ⊂ Ci, and set ω1 = 1+√−32 , ω2 =

√−5.

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Codeword structure

The original codes (before iteration an diagonalization) consist ofcodewords of the form

Xi = λ̃(xi) =[

x1,i+x2,iωi −√−γi(x3,i+x4,iσi(ωi))√−γi(x3,i+x4,iωi) x1,i+x2,iσi(ωi)

],

respectively.

Using these codes as building blocks, the overall 3-relay codes can beconstructed as

Xi =

{Ψηi,3(ατi,θi(X,Y )) = diag

{ηji (ατi,θi(X,Y ))

}2

j=0

∣∣∣∣X,Y ∈ λ̃(Oi)}

The code X1 is 2-group decodable and enjoys a reduction in decodingcomplexity of 50%, from |S|24 to |S|12.

The complexity of the 4-group decodable code X2 is reduced by 75%to |S|6.

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Comparing the performance

No (ST) codes can be found in the literature for N ≥ 3 relays. Thus,we compare the two example codes. (ns = 2, nd = 6).

SNR0 2 4 6 8 10 12 14 16 18 20 22

BLE

R

10-4

10-3

10-2

10-1

100

FD 50FD 75

4-QAM64-QAM

36 / 40

4. Further Applications and Conclusions headlinecolor





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A two-user 2 Tx MIMO-MAC example I

The previous constructions can also yield fast-decodable codes for aK-user MIMO-MAC.

I K = 2 with ns = 2 and a single destination with nd = 4.

I Both transmitters carve their ST codes from the algebra

C =(Q(√−2,√−3, i)/Q(

√−2, i), σ :

√−3 7→ −

√−3,− 2√

5

).

I Let 〈τ : i 7→ −i〉 = Γ(Q(√−2, i)/Q(

√−2)).

I Codewords (for each transmitter) are of the form

Uk = [Xk τ(Xk) ], where for xk ∈ O ⊂ C and θ = 1+√−32 ,

Xk = λ̃(xk) =[

xk,1+xk,2θ −√−γ(xk,3+xk,4σ(θ))√−γ(xk,3+xk,4θ) xk,1+xk,2σ(θ)

].

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A two-user 2 Tx MIMO-MAC example II

The overall transmitted codewords are of the form

X =[X1 τ(X1)X2 τ(X2)

].

This code has the conditional NVD property, and its reduction indecoding complexity is 50%, from |S|32 to |S|16.

SNR

2 4 6 8 10 12 14 16 18 20 22 24

BLE

R

10-4

10-3

10-2

10-1

100

FD, 4-QAM

NFD1, 4-QAM

NFD2, 16-QAM

The two codes chosen from com-parison were adapted to matchthe data rate, and can be foundin [Lu].

H. f. Lu et al. “New space–time code constructions for two-user multiple accesschannels”. In: IEEE J. Sel. Top. Signal Process. 3.6 (2009), pp. 939–957

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Conclusions and future research

I Constructions of fast-decodable space–time codes for the N -relayNAF amplify-and-forward channel.

I Full-diversity and NVD.

I SIMO: 37% reduction in decoding complexity order.Achieves the DMT of the channel.

I MIMO: up to 75% reduction in decoding complexity order.

I Constructions can be adapted to the N-user MIMO-MAC,resulting in fast-decodable codes with CNVD for that setting.

I In future, xtensions to ns > 2, which means dealing withhigher-degree algebras.

I Maybe the new relaxed conditions on fast-decodability facilitatenew constructions.

I More accurate complexity comparison by counting floating pointoperations.

40 / 40


Conclusions and future research

I Constructions of fast-decodable space–time codes for the N -relayNAF amplify-and-forward channel.

I Full-diversity and NVD.

I SIMO: 37% reduction in decoding complexity order.Achieves the DMT of the channel.

I MIMO: up to 75% reduction in decoding complexity order.

I Constructions can be adapted to the N-user MIMO-MAC,resulting in fast-decodable codes with CNVD for that setting.

I In future, xtensions to ns > 2, which means dealing withhigher-degree algebras.

I Maybe the new relaxed conditions on fast-decodability facilitatenew constructions.

I More accurate complexity comparison by counting floating pointoperations.

40 / 40

Kiitos!

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Number-Theoretic Fast-Decodable Space–Time … · Number-Theoretic Fast-Decodable ... {Time Codes from Cyclic Division Algebras ... Number-Theoretic Fast-Decodable Space–Time