Achieving Channel Capacity With Lattice Codes: From Fermat ...

Outline Statement of Coding Problems in Information Theory Lattices and Algebraic Number Theory Coding for Gaussian, Fading and MIMO channels

Achieving Channel Capacity With Lattice Codes:

From Fermat to Shannon

Cong LingImperial College London

[email protected]

May 2, 2016

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1 Statement of Coding Problems in Information Theory

2 Lattices and Algebraic Number Theory

3 Coding for Gaussian, Fading and MIMO channels

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Communications in the presence of noiseSignal vector x = [x1, . . . , xn]T inn-dimensional Euclidean space.

The additive white Gaussian noise(AWGN) channel: y = x + w,where signal power P = E [‖x‖2]/nand noise power = σ2

w .

Shannon capacity (1949)

C =1

2log(1 + ρ)

where signal-to-noise ratio (SNR)ρ = P

σ2w

.

Shannon used random coding, butwe need a concrete code to achievethe capacity.

quadrature amplitude modulation(QAM) constellation

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Coding

Algebraic approach

Hamming code

Reed-Solomon code

BCH code

Algebraic-geometry code

Probabilistic approach

Convolutional code

Turbo code

LDPC code

Polar code

For binary (discrete)-input channels, dream has come true with polarcodes [Arikan’09] and SC-LDPC codes [Jimenez-Felstrom-Zigangirov’99].

However, the question of achieving the capacity of the Gaussianchannel has to be solved with lattice codes.

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Multipath fading in mobile communications

Multipath propagation in urban en-vironment.

Fading is multiplicative noise (largevariation in signal strength)

yt = htxt + wt

Rayleigh fading: channel coefficientht is complex Gaussian.

Time autocorrelation is modelled bya Bessel function (Jakes model)R(τ) = E[hth

∗t+τ ] = J0(2πfdτ)

where fd = (v/c)f is normalizedDoppler frequency shift.

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Slow fading vs. fast fading

0 200 400 600 800 10000

0.5

1

1.5

Time

Fad

ing

Am

plitu

deSlow Fading, Doppler Shift f

dT

s = 0.001

0 200 400 600 800 10000

0.5

1

1.5

2

Time

Fad

ing

Am

plitu

de

Fast Fading, Doppler Shift fdT

s = 0.01

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Models

Slow fading (block fading): The fading process is nearlyconstant (but random) in the duration of a codeword. Weneed (time, frequency etc.) diversity from several independentblocks:

(h1, h1, . . . , h1︸︷︷︸, h2, h2, . . . , h2︸︷︷︸, · · · , hn, hn, . . . , hn︸︷︷︸)The length of each block is known as coherence time T .Ergodicity doesn’t hold due to delay constraint.Capacity C =

∑ni=1 log

(1 + |hi |2ρ

).

Fast fading: The fading coefficients {ht} are nearlyindependent.

In reality, ergodic fading is a more accurate model.Capacity C = EH

[log(1 + |h|2ρ

)].

These represent two extremes of stationary fading.Open question: to design capacity-achieving codes over fadingchannels (wireless systems will operate close to capacity).

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Models



∑ni=1 log

(1 + |hi |2ρ

).



[log(1 + |h|2ρ

)].


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Models



∑ni=1 log

(1 + |hi |2ρ

).



[log(1 + |h|2ρ

)].

These represent two extremes of stationary fading.

Open question: to design capacity-achieving codes over fadingchannels (wireless systems will operate close to capacity).

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Models



∑ni=1 log

(1 + |hi |2ρ

).



[log(1 + |h|2ρ

)].


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Block fading channel

Slow fading is the realistic model in delay-constrained commu-nication systems (4G, 5G...).

Write down the matrix form of the channel Y = HX + W,where channel matrix H = diag[h1, h2, . . . , hn].

Set target capacity

C = log∣∣∣I + ρH†H

∣∣∣ . (1)

The receiver has channel state information (CSI), while thetransmitter doesn’t.

Our goal is to achieve capacity C on all channels such that (1)is true (without even knowing the distribution of H).

This requires a universal code on the compound channel (1),i.e., a collection of channels with the same capacity.

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MIMO channel

Capacity ∝ n, the number of antennas.Channel model Y = HX + W, where H is the n × n channelmatrix, fixed (but random) in coherence time T .Set target capacity


∣∣∣ . (2)

Open question: achieve the capacity of the compound MIMOchannel (2).

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Background on lattices

What are lattices?

Lattices are regular, efficient and near-optimum arrays in theEuclidean space.

The hexagonal lattice A2 and FCC lattice A3 give the densestsphere packings in dimensions 2 and 3.

Breaking news [Viazovska (et al.)’16]: The E8 lattice and Leechlattice are optimum for sphere packing for n = 8 and 24.

Recent years see the revival of this classic area, driven by newapplications, particularly coding and cryptography.

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Why are lattices useful?

Minkowski founded geometric number theory, where lattices areused to solve problems in number theory.Coding for the Gaussian channel is closely related to the spherepacking problem, for which lattices are near-optimum.

Shannon already indicated dense sphere packings to achievechannel capacity (without knowing lattices).

Cryptographers are more interested in the hardness of latticeproblems.

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History of lattice coding and cryptography

1960s: earliest use of lattices incoding.

1984: lattice formulation oftrellis-coded modulation.

1988: first book on lattice cod-ing Sphere Packings, Latticesand Groups.

1992: ideal lattices for Rayleighfading channels.

2004: capacity-achieving latticecodes for Gaussian channels.

2005: lattice-based space-timecodes for MIMO channels.

1982: first use of lattices incryptanalysis.

1996: first crypto scheme basedon hard lattice problems.

2002: first book on latticecrypto Complexity of LatticeProblems published.

2005: learning with errors.

2006: application of ideal lat-tices to crypto.

2009: fully homomorphic en-cryption.

2012: multilinear maps.

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Definition

A lattice Λ ⊂ Rn is defined as

Λ = Λ(B) = {Bx|x ∈ Zn}

where B (n-by-n) is called a basis, or generator matrix. Forexample, the lattice Z2 (aka QAM) has basis B = I2.

May be viewed as the result of a linear transformation appliedto the Zn lattice (cubic lattice).

Euclidean counterpart of a linear code (a linear code is normallydefined in the Hamming space).

The dual lattice Λ∗ has basis (B−1)T . It arises in the Fouriertransform of multi-dimensional signals.

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Fundamental regions

A fundamental region of a lattice is a piece that tiles the Eu-clidean space without any overlap or gap

The volume of a fundamental region is called the fundamentalvolume

V (Λ) = det(Λ) = | det(B)|

Fundamental parallelotope

P = {y | y = Bx, 0 ≤ xi < 1}

Voronoi region: the nearest-neighbor decoding region

V = {y | ‖y‖ < ‖y − x‖, ∀x ∈ Λ}

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Examples of fundamental parallelotope and Voronoi region

Square lattice Z2 Hexagonal lattice A2

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An ensemble of random lattices (Loeliger ensemble)

Consider the following family of q-ary lattices for all matricesA ∈ Zk×n

q ,

Λq(A) = {y ∈ Zn : y = AT s mod q for s ∈ Zk}

These are lattices from Construction A, given the generatormatrix A of the code.

For a lattice vector x,

x mod q = c ∈ C

for a linear (n, k) code C ⊂ Znq.

Construction A is an important bridge between lattice theoryand coding theory.

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Minkowski-Hlawka Theorem

Loeliger ensemble is discrete version of classic Minkowski-Hlawka-Siegel emsemble:

L = {Λ(B) : B ∈ SLn(R)/SLn(Z)}There exist dense lattices in Loeliger ensemble as n, q → ∞[Lolieger’97]

λ1(Λ) ≈√

n

2πeV 1/n(Λ)

Its proof is based on Shannon’s random coding.Rogers’ proof of Minkowski-Hlawka Theorem is a special casewith k = 1.

Good lattices can be generated from random codes.Good for coding.Good for quantization.Good for secrecy....

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Variations

Similarly, Construction A may be defined with the parity-checkmatrix:

Λ⊥q (A) = {y ∈ Zn : Ay = 0 mod q}

Some special lattices/generalizations

Cyclic codes ⇒ cyclic latticesNegacyclic codes ⇒ negacyclic latticesDouble-circulant codes: aka NTRU in crypto; quasi-cycliccodesModulo a multi-dimensional lattice (D4, E8), ideal q of anumber field...

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Quest for structured latticesConstruction A

Good news: dense lattices (for coding [Loeliger’97]); hard todecode (for crypto [Regev’05]).Bad news: hard to decode (for coding); inefficient (for both)a.

aBeing efficient means quasi-linear complexity; n is several hundreds tothousands in practice.

In coding: (the study of transmitting information)Gaussian channels:

Classic approach: dense lattices.Practical approach: Constructions A, D etc. from good codes.

Fading channels: ideal lattices from algebraic number fields.MIMO channels: division algebras over number fields.

In crypto: (the study of hiding information)Cyclic lattices, ideal lattices.More efficient than general lattices.New functionalities: homomorphic encryption, codeobfuscation... 19 / 47


Algebraic number theory


Fermat’s Last Theorem (1637): When n > 2,

xn + yn = zn

has no nontrivial solutions x , y , z ∈ Z.

It was in the Guinness Book of World Records for “most difficultmathematical problems”.

Historically gave rise to algebraic number theory:

(xp + yp) =

p−1∏i=0

(x + ζpy)

Kummer proved the theorem for all regular primes (p - hp of acyclotomic number field).

Finally settled by Andrew Wiles in 1994, 3.5 centuries later.

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Number fields

A number field K is a finite field extension of Q, i.e., a fieldwhich is a Q-vector space of finite dimensions. The dimension[K : Q] is called the degree of K .

Any number field can be built by adding a primitive element θto Q, i.e., K = Q(θ) (in fact, θ is an algebraic integer).

An algebraic integer in a number field K is an element α ∈ Kwhich is a root of a monic irreducible polynomial with integercoefficients. Such a polynomial is called the minimum polyno-mial of α.

Example: Q(√

2) = {x + y√

2|x , y ∈ Q} is a number field ofdegree 2, i.e., a quadratic field.

Example: If ζm is a primitive mth root of unity, the numberfield Q(ζm) is called a cyclotomic field.

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Ring of integers

The ring of integers OK of a number field K is the set of allalgebraic integers of K .

Example: Z[√

2] = {x + y√

2|x , y ∈ Z} is the ring of integersof Q(

√2).

Example: For the mth cyclotomic number field Q(ζm) of degreen = ϕ(m), the ring of integers is given by

Z[ζm] = Z + Zζm + . . .+ Zζn−1m∼= Z[X ]/〈Φm(X )〉.

There exists an integral basis {ωi}ni=1 of K such that any ele-ment of OK can be uniquely written as

∑ni=1 aiωi with ai ∈ Z

for all i .

We can get an algebraic lattice from OK .

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Canonical embedding

Let θi for i = 1, . . . , n be the distinct roots of the minimumpolynomial of θ. There are exactly n embeddings σi : K → C,defined by σi (θ) = θi , for i = 1, . . . , n.

When we apply the embedding σi to an arbitrary element x ofK , x =

∑nk=1 akθ

k , ak ∈ Q, we get

σi (x) = σi (n∑

k=1

akθk) =

n∑k=1

σi (ak)σi (θ)k =n∑

k=1

akθki

Let r1 be the number of embeddings with image in R, and 2r2the number of embeddings with image in C so that r1+2r2 = n.

Canonical (Minkowski) embedding

σ(x) = (σ1(x), . . . , σr1(x),R(σr1+1(x)), . . . ,I(σr1+r2(x))) ∈ Rn.

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From OK to lattice

If we take the ring of integers OK , we obtain a lattice withcanonical embedding.

Let {ωi}ni=1 be an integral basis of K . The n vectors vi =σ(ωi ) ∈ Rn form a basis of an algebraic lattice Λ = Λ(OK ) =σ(OK ), whose generator matrix is given by

M =

σ1(ω1) · · · σr2(ω1) Rσr1+1(ω1) · · · Iσr1+r2(ω1)σ1(ω2) · · · σr2(ω2) Rσr1+1(ω2) · · · Iσr1+r2(ω2)

......

......

......

σ1(ωn) · · · σr2(ωn) Rσr1+1(ωn) · · · Iσr1+r2(ωn)

.

We can get more lattices Λ′ ⊂ Λ from ideals I ⊆ OK , whichare called ideal lattices.

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Gaussian Channel

Lattice coding

A lattice code is a code constructed from a lattice in the Eu-clidean space. Thus, it is naturally suited to a Gaussian channel.Since a lattice is infinite, shaping is needed to obtain a code ofgiven rate. The common practice is to apply a finite shapingregion (cubic, spherical, or Voronoi).

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Gaussian Channel

AWGN-good lattices

The issues of shaping and coding are largely separable.

Consider (infinite) lattice coding over the AWGN channel withnoise variance σ2

w .

For an n-dimensional lattice Λ, define the volume-to-noise ratio(VNR) by

γΛ(σw ) ,(V (Λ))

2n

σ2w

The error probability is given by Pe = P{Wn /∈ V(Λ)}.A sequence of lattices Λ(n) of increasing dimension n is AWGN-good if for a fixed VNR γΛ(σw ) > 2πe, Pe vanishes in n[Poltyrev’94].

This is the best possible performance, achieved only if theVoronoi region is approximately a sphere.

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Gaussian Channel

Constructions from number fields

The connection between coding and dense sphere packing iswell known [Conway-Sloane’93].

Lattices from cyclotomic fields [Craig’78]: Let p = n + 1 bea prime. Take a principal ideal I = ((1 − ζp)m) in Q(ζp), forsome m. I yields a lattice under canonical embedding.

Lattices from class field towers [Martinet’78]: Embedding ofOKi

in the tower; densest lattices having been constructed.

Remark

Number field constructions yield dense lattices, but have notachieved the Minkowski-Hlawka bound.

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Gaussian Channel

Constructions from codes

Constructions A, B, C, D... [Leech-Sloane’71].

Existence of AWGN-good lattices from Construction A [Loeliger’97].

Shannon-theoretic construction [Forney et al.’00]: A latticefrom Construction A is AWGN-good if the code achieves ca-pacity of the mod-q channel.

Remark

Forney et al.’s construction yields AWGN-good lattices, but are notnecessarily dense.

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Gaussian Channel

Achieving capacity with lattice codes

Lattice codes achieve the 12 log(1 + ρ) capacity of the Gaussian

channel.

This also forms the basis of the recent surge in applications tonetwork information theory.

Erez and Zamir’s scheme [2004]

Voronoi shaping: the coarse lattice is good for quantization.It also requires dithering.The existence proof is again based on random lattices.

Polar lattice

Gaussian shaping: Applying a discrete Gaussian distributionover an AWGN-good lattice [Ling-Belfiore’14].Explicit construction from polar codes, O(n log n) decodingcomplexity [Yan-Liu-Ling-Wu’15].

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Fading Channel

Coding for fading channel

Good Codes for the Gaussian channel usually have rather poorperformance in the fading channel.

The construction of good lattice codes for the fading channelexploits algebraic number theory [Belfiore et al. 1990s].

A powerful tool is ideal theory for the rings of algebraic integers,leading to the construction of ideal lattice codes.

However, capacity-achieving codes for fading channels are stillunavailable1.

Record is a constant gap to capacity [Ordentlich-Erez’13, Luzzi-Vehkalahti’15].

1In the special case of i.i.d fading, capacity is acheived with polar lattices[Liu-Ling’16]

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Fading Channel

Work in 1990s

Consider the fast (i.i.d.) Rayleigh fading channel

yi = hixi + wi

where (x1, . . . , xn) = x ∈ Rn is the codeword, hi ’s are i.i.d.fading coefficients of the Rayleigh distribution.

Pairwise error probability

P(x→ x) ≤ 1

2

∏i :xi 6=xi

8σ2w

(xi − xi )2=

1

2

(8σ2w )I∏

i :xi 6=xi(xi − xi )2

if the two codewords differ in I positions.Design criteria

Maximize the diversity order min{I} = minx 6=x |{i : xi 6= xi}|.Full diversity order n is desired.Maximize the product distance:dp,min = minx 6=x

∏i :xi 6=xi

|xi − xi |.31 / 47


Fading Channel

Ideal lattice code

Now, suppose an ideal lattice Λ built from ideal I ⊆ OK isused as the coding lattice.

By the union bound and geometric uniformity, error probability

Pe ≤∑x∈Λ\0

P(0→ x) =∑x∈Λ\0

1

2

(8σ2w )Ix∏

i :xi 6=0 |xi |2

where Ix is the number of nonzero elements (the sum is takeover a shaping region).

Design criteria rephrased (Oggier, Viterbo’04):

Maximize the diversity order min{I} = minx∈Λ\0 |{i : xi 6= 0}|.K should be totally real to achieve full diversity n.The minimum norm Nmin = minx 6=0,x∈I |N(x)| should be

maximized (recall algebraic norm N(x) ,∏n

i=1 σi (x)).

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Fading Channel

Capacity?

Our goal for block fading channels is to achieve capacity ofcompound channel (with diagonal H)


∣∣∣ .

We need coding over time. Recall the system model

Y︸︷︷︸n×T

= H︸︷︷︸n×n

X︸︷︷︸n×T

+ W︸︷︷︸n×T

Vectorizing this equation, we obtain

y︸︷︷︸nT×1

= H︸︷︷︸nT×nT

x︸︷︷︸nT×1

+ w︸︷︷︸nT×1

where H = IT ⊗H.

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Fading Channel

Capacity?



∣∣∣ .We need coding over time. Recall the system model

Y︸︷︷︸n×T

= H︸︷︷︸n×n

X︸︷︷︸n×T

+ W︸︷︷︸n×T


y︸︷︷︸nT×1


x︸︷︷︸nT×1

+ w︸︷︷︸nT×1

where H = IT ⊗H.

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Fading Channel

Capacity?



∣∣∣ .We need coding over time. Recall the system model

Y︸︷︷︸n×T

= H︸︷︷︸n×n

X︸︷︷︸n×T

+ W︸︷︷︸n×T


y︸︷︷︸nT×1


x︸︷︷︸nT×1

+ w︸︷︷︸nT×1

where H = IT ⊗H.

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Fading Channel

Fading-good lattices

Now we design a lattice Λ ⊂ CnT so that x ∈ Λ.

With Gaussian shaping, the problem boils down to finding alattice that is good for block fading.

Fading-good lattices [Campello-Ling-Belfiore’16]

We say that a sequence of lattices Λ of increasing dimension nT isuniversally good for the block-fading channel if for any VNRγ(IT⊗H)Λ(σw ) > 2πe and all (absolute) H s.t. |H| = D,Pe(Λ,H)→ 0 as T →∞.

If H = I, the problem reduces to that of AWGN-good lattices.

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Fading Channel







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Fading Channel







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Fading Channel

Generalized Construction A

We resort to generalized Construction A over OK .

Generalized Construction A [Kositwattanarerk-Ong-Oggier’13]

Let K/Q(i) be a relative extension of degree n.Let p ⊂ OK be a prime ideal above p with norm p`. ThenOK/p ' Fp` .

The OK -lattice Λ associated to a linear code C ⊂ FTp`

is defined as:

Λ = C + pT .

It reduces to usual Construction A: Λ = C+ pT when K = Q.

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Fading Channel






is defined as:

Λ = C + pT .


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Fading Channel






is defined as:

Λ = C + pT .


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Fading Channel

Achieving capacity

Generalized Construction A is good for fading channels

With number fields, generalized Construction A are good for block fading[Campello-Ling-Belfiore’16].The existence of a universal lattice can be proven by Minkowski-Hlawka,i.e., averaging over random codes C (with p →∞).

Thanks to the unit group, the set of quantized channels is always

compact (the unit group “absorbs” the channel).

With Gaussian shaping, capacity of compound fading channelsis achieved.

However, there is a large gap between theory (given above) andstate of the art [Ordentlich-Erez’13, Luzzi-Vehkalahti’15].

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Fading Channel

Achieving capacity







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Fading Channel

Achieving capacity







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Fading Channel

Universality (for T = 1)

In general, there is no guarantee that a faded lattice still hasgood minimum distance.

Nevertheless, an ideal lattices I is “incompressible” [Luzzi-Vehkalahti’15]: the minimum Euclidean distance of faded lat-tice HI

minH:|H|=D

d2min(HI) = min

H:|H|=Dmin

x∈I,x6=0‖Hx‖2

= minx∈I,x6=0

nD2/n(x1 · · · xn)2/n

= nD2/nN(I)2/n

which follows from AM-GM inequality.

However, dmin = 0 for usual mod-q lattices.

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Fading Channel




minH:|H|=D

d2min(HI) = min

H:|H|=Dmin

x∈I,x6=0‖Hx‖2

= minx∈I,x6=0

nD2/n(x1 · · · xn)2/n

= nD2/nN(I)2/n



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Fading Channel




minH:|H|=D

d2min(HI) = min

H:|H|=Dmin

x∈I,x6=0‖Hx‖2

= minx∈I,x6=0

nD2/n(x1 · · · xn)2/n

= nD2/nN(I)2/n



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MIMO Channel

Error probability of MIMO

Recall channel model Y︸︷︷︸n×T

= H︸︷︷︸n×n

X︸︷︷︸n×T

+ W︸︷︷︸n×T

.

For a linear space-time code S, consider pairwise error proba-bility

P(0→ X) = EH

[Q

(‖HX‖F√

2σw

)](3)

For Rayleigh fading,

P(0→ X) ≤

∣∣∣∣∣I +XX†

4σ2w

∣∣∣∣∣−n

(4)

If codeword matrix X has full rank, then high-SNR behavior

P(0→ X) ≤ |XX†|−n(

1

4σ2w

)−n2

(5)

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MIMO Channel

Non-vanishing determinant (NVD)

Define∆ = inf

0 6=X∈S|XX†| (6)

Non-vanishing ∆ > 0 implies full diversity, in fact, optimumDMT (diveristy-multiplexing gains tradeoff) of the space-timecode S.

Larger ∆ gives larger coding gain.

Many approaches were tried in 2000’s, but ∆ → 0 as constel-lation grows for most of them.

Solution: construct a lattice code from division algebra [Sethuraman-Rajan’02] over a number field [Belfiore-Rekaya’03].

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MIMO Channel

Division algebra

Definition

Let A be a ring and denote by A∗ the set of invertible elements ofA for multiplication. If A∗ = A\{0}, then A is referred to as adivision algebra.

Hamilton’s quaternions

Let {1, i , j , k} be a basis for a vector space of dimension 4 over R,which satisfy the relations i2 = −1, j2 = −1, k2 = −1 andk = ij = −ji . Hamilton’s quaternions are defined as the set

H = {x + yi + zj + wk | x , y , z ,w ∈ R}.

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MIMO Channel

The key link

Rewrite a quaternion q = x + yi + zj + wk as

q = (x + yi) + j(z − wi) = α + jβ

where α = x + yi and β = z − wi .

It matrix representation

q ⇐⇒ X =

(α −ββ α

).

Check

|X| = |α|2 + |β|2 = N(q) > 0 if X 6= 0

The norm N(q) = 0 if and only if q = 0.

This is the famous Alamouti code with full diversity [Alam-outi’98]. It’s used in DVB, WiFi and 4G.

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MIMO Channel

Cyclic algebra

Definition

Let L/K be a Galois extension of degree n whose Galois group iscyclic with generator σ. Choose an element 0 6= γ ∈ K . A cyclicalgebra A = (L/K , σ, γ) is defined as the direct sum

A = L⊕ eL⊕ · · · ⊕ en−1L

whereen = γ and λe = eσ(λ) λ ∈ L.

The cyclic algebra may be viewed as a vector space over L, i.e., anelement x ∈ A is written as

x = x0 + ex1 + · · ·+ en−1xn−1 for xi ∈ L.

The rule λe = eσ(λ) defines multiplication.42 / 47


MIMO Channel

Matrix representation

An element x ∈ Ax = x0 + ex1 + · · ·+ en−1xn−1 for xi ∈ L

can be represented by

x0 γσ(xn−1) γσ2(xn−2) . . . γσn−1(x1)x1 σ(x0) γσ2(xn−1) . . . γσn−1(x2)...

......

xn−2 σ(xn−3) σ2(xn−4) . . . γσn−1(xn−1)xn−1 σ(xn−2) σ2(xn−3) . . . σn−1(x0)

. (7)

Cyclic division algebra [Oggier-Belfiore-Viterbo’07]

If 0 6= γ, γ2, . . . , γn−1 ∈ K are not a norm of some element of L,then A = (L/K , σ, γ) is a cyclic division algebra.

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MIMO Channel

Golden code (n = 2, optional in WiMAX)

In general, a space-time code is an order of A. Then |X| is reducednorm and ∆ > 0 naturally.If n = 2, consider the cyclic division algebra [Belfiore-Rekaya-Viterbo’05]

A = (L = Q(i ,√

5)/Q(i), σ, i)

where σ :√

5 7→ −√

5. The ring of integers OL is given by

OL = {a + bθ | a, b ∈ Z[i ]}

where θ = 1+√

52 . A codeword of the Golden code is of the form

X =

(a + bθ c + dθ

i(c + dσ(θ)) a + bσ(θ)

)where a, b, c , d ∈ Z[i ].

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MIMO Channel

Construction A from division algebras

Generalized Construction A[Vehkalahti-Kositwattanarerk-Oggier’14]

Let Λ be the natural order of cyclic division algebra A.Take a two-sided ideal J of Λ and consider the quotient ring Λ/J .Define a reduction β : Λ→ Λ/J .For a linear code C over Λ/J , β−1(C) is a lattice (in Cn2T ).

Λ/J can be a matrix ring, skew polynomial ring...

Nevertheless still possible to prove Minkowski-Hlawka usingcodes over rings [Campello-Ling-Belfiore’16].

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MIMO Channel

NVD implies universality (for T = 1)

Again, Λ with NVD ∆ > 0 is “incompressible” [Luzzi-Vehkalahti’15]:the minimum Euclidean distance of faded lattice HΛ

minH:|H|=D

d2min(HΛ) = min

H:|H|=Dmin

X∈Λ,X 6=0‖HX‖2

F

= minX∈Λ,X 6=0

nD2/n|X|2/n

= nD2/n∆2/n

which follows from Hadamard’s inequality and AM-GM inequal-ity.

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MIMO Channel

Concluding remarks

The coding problem for Gaussian channels has been solved.

Algebraic number theory is an indispensable tool to design mod-ern coding systems over fading/MIMO channels.

Achieve capacity of compound fading channels requires acombination of number theory and coding theory.These can be viewed as concatenated codes (inner code isideal/order; outer code is a code).Extension to ergodic fading is possible.

Emerging applications to multi-user networks:

Compute-and-forwardInterference alignment

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Achieving Channel Capacity With Lattice Codes: From Fermat ...

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