On Codes and Lattices

Sueli I. R. Costa, University of Campinassueli@ime.unicamp.br

Collaboration: Antonio Campello, University of Campinas

SPCodingSchoolJanuary 20-21, 2015

Campinas, Brazil

Outline

I Intro/History

I Initial Definitions - Invariants

I Parameters

I Important Lattices

I Computational Problems/Lattices in Cryptography

I Lattice Constructions

I Lattice Coding for Gaussian and Fading Channels

I Lattices and Spherical Codes

General References

I J. H. Conway, N. J. A. Sloane, Sphere Packing, Lattices andGroups, Springer-Verlag, 1998

I Zamir R., Lattices are Everyhwere, ITA 2009 (IEEExplore)

I Zamir R., Lattice Coding for Signals and Networks, 2014

I Regev-Micciancio Lattice Based-Crypto in Post-QuantumCryptography, Springer, 2009.

I C. Lavor, M. Muniz, S. Siqueira, S. I. R. Costa, UmaIntroducao a Teoria de Codigos, Notas em MatematicaAplicada, SBMAC, 2006 (in Portuguese, freely available inhttp://www.sbmac.org.br/p_notas_titulos.php

Related Short Courses

I “Lattice Coding for Signals and Networks” by Ram Zamir

I “Explicit Lattice Constructions: From Codes to NumberFields” by Frederique Oggier and Jean-Claude Belfiore

Definition

I Given m linearly independent vectors bbb1, . . . ,bbbm ∈ Rn, alattice with basis {bbb1, . . . ,bbbm} is defined as

Λ = {u1bbb1 + . . .+ umbbbm : u1, . . . , um ∈ Z} .

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

Z2 A2 (Hexagonal)

Definition

I Z3 lattice (integer coordinates)

I FCC (Face-centered cubic)

A Brief History

I 18th century: mathematicians such as Gauss and Lagrangeuse lattices in number theory; 19th century Minkowski greatlyadvances the study of lattices in his Geometry of numbers.(***) Fields Medal 2014 - Manjul Bhargava

I Applications to Coding and Information TheoryI [1975] De Buda - Lattice Codes for the Gaussian ChannelI [1987-1992] Calderbank-Sloane, D. Forney - Treliis Codes,

Coset Codes, Geometrically Uniform CodesI [1992-2004] Loeliger (Asymptotically Good Lattices), Zamir

(Quantizers, Wyner-Ziv), Boutros-Viterbo-Rastello-Belfiore(Fading Channels), Zamir-Shamai-Erez,

I [2004-2005] Erez-Zamir and Erez-Litsyn-Zamir - LatticesWhich Are Good For (Almost) Everything (AWGN, Coding,Quantization, Packing, Covering)

I Subsequent works on: side information problems, jointsource-channel coding, spherical codes, ...

I General Reference: R.Zamir: Lattices are Everywhere (ITA2009).

A Brief History

I CryptographyI Early 80s: LLL basis reduction. Factoring polynomials and

breaking cryptosystems.I 90s: Ajtai one-way function (hash) based on worst-case

hardness conjectures; Dwork (public-key encryption - theoreticinterest); Hoffestin, Pipher and Silverman (“efficient”, butlacks security).

I 1996: P. Short (factorization on a quantum computer)I 2000s: Regev and Micciancio, most stronger security, improved

efficiencyI 2007 - present (Post-Quantum Cryptography): rich toolbox of

lattice based cryptographic constructionsI References: C. Peikert, Lattices and Cryptography (Lecture

Notes), Regev-Micciancio, Lattice Based-Crypto inPost-Quantum Cryptography, 2009.

Initial Definitions

I Given m linearly independent vectors bbb1, . . . ,bbbm ∈ Rn, alattice with basis {bbb1, . . . ,bbbm} is defined as

Λ = {u1bbb1 + . . .+ umbbbm : u1, . . . , um ∈ Z} .

I If m = n, we say that Λ is a full-rank lattice.

I Generator matrix: Matrix B whose columns* are bbbi :

B =(bbb1 . . . bbbm

)

I Λ = Λ(B) = {Buuu : uuu ∈ Zm}I Gram Matrix: G = BBt .

Initial Definitions

Discrete additive subgroup:

I xxx ,yyy ∈ Λ⇒ −xxx and xxx + yyy ∈ Λ

I There exists ε such that the balls of radius ε, B(ε) + xxx , xxx ∈ Λare disjoint.

Equivalent Characterization

A set in Rn is a lattice if, and only if, it is a discrete additivesubgroup of Rn.

Proof: ⇒ (“if”) all lattices are discrete additive subgroups of Rn:easy.⇐ (“only if”): exercise. (Hint: how to construct a basis?)

Examples (R2)

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

(1 1/2

0√

142

) (1 00 1

) (1 1/2

0√

32

)

Examples (R3)

I ∆(Z3) = 0.5236

I Body-centered cubic lattice ∆(BCC ) = 0.6802

Examples (R3)

I Face-centered cubic lattice ∆(FCC ) = 0.7405

Initial DefinitionsA lattice has infinitely many bases.

Change of Basis

Λ(B) = Λ(B) iff B = UB, where U is an unimodular matrix (i.e.,integer entries and determinant one).

Sketch of the Proof: Write the columns of B as integercombinations of the columns of B and vice-versa.

Example 1

U =

(−312 479−71 109

)is unimodular. Hence the lattice generated

by U is the same as the one generated by the identity matrix I2×2

(the Z2 lattice).

Example 2

B =

(−695

21067

2

−12

(71√

3)

109√

32

)is another generator matrix for the

hexagonal lattice.

Two bases for the Hexagonal Lattice

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H0,0L H1,0L

1

2J1, 3 N

1

2J3, 3 N

1

2J7, 3 3 N

B =

(1 1/2

0√

3/2

)(blue) and B =

(3/2 7/2√3/2 3

√3/2

)(red)

Some bases are better than others. Good basis: Vectors“sufficiently orthogonal” with ”small norm”

Two bases for the hexagonal lattice

Two different bases: same area of the parallelotope.

Initial Definitions

I The determinant of a lattice is defined as

det Λ = detG = detBBt

I It corresponds to the squared volume of a fundamentalparallelotope

P(B) = {α1bbb1 + . . .+ αmbbbm : 0 ≤ αi < 1, i = 1, . . . ,m}

I The determinant is also teh volume of a Voronoi Region

VΛ(xxx) = {yyy ∈ span(B) : ‖xxx − yyy‖2 ≤ ‖zzz − yyy‖2 , for all zzz ∈ Λ} .

Z2

Z2

Z2

Hexagonal

Hexagonal

Hexagonal

Arbitrary Lattice

Arbitrary Lattice

Arbitrary Lattice

Parameters

I Packing: Density

I Covering: Thickness

I Kissing Number

I Quantization: Normalized Second Moment

Packing Density

I Minimum norm:λ(Λ) = min

06=xxx∈Λ‖xxx‖2

I Packing radius:ρ(Λ) = λ(Λ)/2.

I Packing density:

∆(Λ) =vol B2(ρ)√

det Λ,

where vol B2(ρ) is the volume of an euclidean sphere of radiusρ in Rn. (

vol B2(1) =πn/2

(n/2)!

)

Packing density

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

∆ = 0.629 ∆ = 0.785 ∆ = 0.906

Packing

vol B2(ρ) = ρn (vol B2(ρ))

I Center densityδ(Λ) = ∆(Λ)/vol B2(1)

I Hermite Parameter:

γ(Λ) =λ(Λ)2

det(Λ)2/n= 4δ2/n.

The maximum Hermite parameter among n-dimensional lattices isthe Hermite Constant γn. It is known (but not trivial to show)that γn grows linearly with n.

Covering

Let B2(ρ) be a ball of radius ρ. We say that the set

Λ + B2(ρ) =⋃

xxx∈Λ

xxx + B2(ρ)

is a covering if Rn = Λ + B2(ρ) (i.e., each point in space is coveredby at least one sphere).

Covering

I Covering radius:

ρcov(Λ) = min {ρ : Λ + B2(ρ) is a covering }

I Thickness Θ(Λ)

Θ(Λ) =vol B2(ρcov)

det Λ.

I Normalized Thickness (analogous to center density):

θ(Λ) =ρncov

det Λ.

Kissing Number

I How many balls can be arranged so that they all just touch(or “kiss”) another central ball of the same size?

I R3: Gregory versus Newton . Correct answer: 12.

I Rn: not known in general

I Kissing number τ(Λ) of a lattice: number of “closestneighbors” (number of spheres centered at lattice points thattouch a central ball centered at a lattice point).

Kissing Number

α = π3

α = π3

d = 1

Kissing Number

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

τ = 4 τ = 2 τ = 6

Quantization: Normalized Second Moment

The second moment per dimension of a lattice is the averageenergy of points uniformly distributed in a Voronoi Cell:

σ(Λ)2 =1

n

1

V (Λ)

∫

V‖xxx‖2 dxxx .

To make this quantity dimensionless it is usual to consider theNSM (Normalized Second Moment)

G (Λ) =σ(Λ)2

V 2/n(Λ).

The lattice quantizer problem is to find the n-dimensional lattice Λthat minimizes G (Λ). For lattices achieving asymptotic bounds forG (Λ), see the tutorial by R. Zamir.

Equivalent Lattices

Two lattices are said to be equivalent if one can be obtained by theother through an orthogonal transformation and a change of scale.Two equivalent lattices have the same packing density, coveringthickness, kissing number and normalized second moment.

Important Lattices

I The cubic lattice Zn. Generator matrix: Identity n× n (or anyunimodular matrix).

I detZn = 1.

I Center density: δ(Zn) = 1/2n

I Normalized Thickness: (√n/2)n.

I Kissing Number: τ = 2n.

Important Lattices

I The checkerboard lattice Dn:

Dn = {(x1, . . . , xn) ∈ Zn : x1 + . . .+ xn is even} .

Generator matrix:

2 0 0 . . . 0 01 −1 0 . . . 0 00 −1 1 . . . 0 0...

......

. . ....

...0 0 0 . . . −1 1

,

Important Lattices

I The lattice An (in Rn+1)

An ={

(x1, . . . , xn+1) ∈ Zn+1 : x1 + . . . xn+1 = 0}

= Zn+1 ∩ (1, . . . , 1)⊥.

Generator matrix:

−1 1 0 . . . 0 00 −1 1 . . . 0 0...

......

. . ....

...0 0 0 . . . −1 1

.

Important LatticesThe dual of a lattice: Given Λ ⊂ Rn, the dual is defined as

Λ∗ = {(x1, . . . , xn) ∈ span(B) : 〈xxx ,yyy〉 ∈ Z for all yyy ∈ Λ} .

Example: If Λ is generated by

(2 00 1

), Λ∗ is generated by

(1/2 0

0 1

).

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H0, 0L H2, 0L

H1, 0L

}

1 � 2

Unimodular Lattices

I A lattice is called integral if 〈xxx ,yyy〉 ∈ Z for any xxx ,yyy ∈ Λ (or,equivalently, Λ ⊂ Λ∗).

I A lattice is called unimodular if Λ = Λ∗ (analogous toself-dual codes).

I A lattice is even if 〈xxx ,xxx〉 is even for all xxx ∈ Rn.

TheoremIf Λ ⊂ Rn is an even unimodular lattice, then n is a multiple of 8

For a proof see, e.g., Ebeling, Lattices and Codes.Example: E8, Λ24,...

The lattices E6,E7,E8

The lattice E8 is the unique even unimodular lattice (up toequivalence) in dimension 8, generated by

2 0 0 0 0 0 0 0−1 1 0 0 0 0 0 00 −1 1 0 0 0 0 00 0 −1 1 0 0 0 00 0 0 −1 1 0 0 00 0 0 0 −1 1 0 00 0 0 0 0 −1 1 012

12

12

12

12

12

12

12

. (1)

The lattices E6 and E7 are defined as

E7 = E8 ∩ (1, . . . , 1)⊥ e E6 = E8 ∩ H⊥,

where H is the two-dimensional vector space generated by(1, 0, . . . , 0, 1) and (1/2, . . . , 1/2).

The Leech Lattice Λ24

A scaled version: 2√

2Λ24 generator matrix

8 4 4 4 4 4 4 2 4 4 4 2 4 2 2 2 4 2 2 2 0 0 0 −30 4 0 0 0 0 0 2 0 0 0 2 0 2 0 0 0 0 0 2 2 0 0 10 0 4 0 0 0 0 2 0 0 0 2 0 0 2 0 0 2 0 0 2 0 0 10 0 0 4 0 0 0 2 0 0 0 2 0 0 0 2 0 0 2 0 2 0 0 10 0 0 0 4 0 0 2 0 0 0 0 0 2 2 2 0 2 2 2 2 0 0 10 0 0 0 0 4 0 2 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 10 0 0 0 0 0 4 2 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 10 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 10 0 0 0 0 0 0 0 4 0 0 2 0 2 2 2 0 2 2 2 2 2 2 10 0 0 0 0 0 0 0 0 4 0 2 0 2 0 0 0 2 0 0 0 2 0 10 0 0 0 0 0 0 0 0 0 4 2 0 0 2 0 0 0 2 0 0 0 2 10 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 4 2 2 2 0 0 0 0 2 2 2 10 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 2 2 2 2 2 2 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Leech Lattice Modem

“A Leech Lattice Modem”, by Gordon R. Lang and Frederick M.Longstaff (1989)

Best Known Lattices

A2 = A∗2A3 = D3 = FCCA∗3 = D∗3 = BCCE8 = E ∗8Λ24 = Λ∗24.

Bounds/Minkowski Hlawka

Minkowski Bound on λ(Λ)

For any n-dimensional lattice Λ,

λ(Λ) ≤√n(det Λ)1/n.

Minkowski-Hlawka TheoremThere exists an n-dimensional lattice such that

∆(Λ) ≥ ζ(n)

2n−1,

where ζ(n) = 1 + 12n + 1

3n + . . . is the Riemman Zeta Function.Proof: Random ensembles of lattices (see R. Zamir’s tutorial...)

Corollary

The Hermite constant γn grows linearly with n.

Computational Problems

I Shortest Vector Problem Given a lattice Λ find its shortestvector.

I Closest Vector Problem Given a lattice Λ ∈ Rn and a vectorr ∈ Rn find the closest lattice point to r .

Computationally hard (NP-hard).

Cryptographic primitives based on the hardness of CVP, SVP andrelated decision problems.

Ideal Lattices - Construction A

SVP Challenge - Ideal Lattices ChallengesReference: Micciancio D. and Regev O., Lattice BasedCryptography in Post-Quantum Cryptography (Berstein,Buchman, Johannes, Dahmen, Erik (Eds.)

Lattices in Crypto

Intuitive Public Key Cryptosystem

I Reminiscent of McEliece code-based cryptosystem

I Alice: Private key: B (a “good” generator matrix Λ).Public key: H (a “bad” generator matrix for Λ)

I Encryption: Given a message x , pick a random short errorvector e, so that the encrypted data is y = Hx + e.

I Decryption is easy given (B, y) but is hard given (H, y).

I Various adaptations/definitions of good/bad generatormatrices (e.g. HNF).

Lattices in Secrecy

Another applications of lattices to security: Lattices for theGaussian Wiretap Channel - see short course by J.-C. Belfiore andF. Oggier

Important parameters: Lattice Coset Codes and Theta Series

Further Definitions

Let q = eπiz . The theta series of a lattice is defined as

ΘΛ(z) =∑

xxx∈Λ

q〈xxx ,xxx〉.

It describes the “lattice point distribution”, and may be thought asa formal series in q. If Λ is integral:

ΘΛ(z) =∑

xxx∈Λ

q〈xxx ,xxx〉 =∞∑

m=0

Nmqm,

where Nm is the number of lattice points with squared norm equalto m. Also notice that

ΘΛ(z) = 1 + τ(Λ)qλ(Λ)2+ higher order terms.

For many applications this approximation is enough. For others, itis not...

Basic Concepts/Properties

I If Λ ⊂ Λ is also a lattice, we call it a sublattice.

I The quotient Λ/Λ defines equivalent classes, so that x1 and x2

are in the same class iff x1 − x2 ∈ Λ.

I The cardinality of the quotient is given by

∣∣∣∣Λ

Λ

∣∣∣∣ =det Λ

det Λ.

I Special sublattices: Λ = lattices with orthogonal basis.

Construction ALattices from q-ary codes. Natural way to “lift” a code to alattice. Let C ⊂ Zn

q be q q-ary code.

Consider the mapping

φ : Zn → Znq

(x1, . . . , xn)→ (x1, . . . , xn),

where xi = xi mod q. ΛC = φ−1(C).

I ΛC is a lattice iff C is a linear code (ΛC is called q-ary latticeor a modulo-q lattice).

I qZn ⊂ Λ ⊂ Zn.

Alternative characterization:

ΛC = {xxx ∈ Zn : xxx mod q ∈ C} .

ΛC = C + qZn.

Construction A

C = {(0, 0), (1, 3), (2, 6), (3, 2), (4, 5), (5, 1), (6, 4)} ⊆ Z27

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Construction A

Properties

(i) The number of codewords of C is given by

|C| =

∣∣∣∣ΛA(C)

qZn

∣∣∣∣ =qn

det(ΛA(C)).

(ii) If C is generated by (I M)t , then

B =

(Ik×k 0k×(n−k)

M(n−k)×k qI(n−k)×(n−k)

)(2)

is a generator matrix for ΛA(C).

(iii) Every lattice Λ ⊆ Zn is q-ary, for 1 ≤ q ≤ det Λ.

Binary Construction A

For binary codes, the following holds:

Λ(ΛC )2 = min {4, dHam(C )} ,where dHam(C ) is the minimum Hamming distance of C . It is alsousual to consider the scaled version 1√

2ΛC because of the following

Theorem

I C is self-dual if and only if 1√2

ΛC is unimodular.

I C is doubly even (every codeword has weight divisible by 4) ifand only if 1√

2ΛC is even.

Binary Construction A

As a simple example consider the parity-check code

C = {(x1, . . . , xn) ∈ Zn2 : x1 + . . .+ xn = 0} .

ΛC = {(x1, . . . , xn) ∈ Zn : x1 + . . .+ xn is even} = Dn.

Since dHam(C ) = 2, λ(Λ) = 2. Also detDn = 2n/|C | = 2.

A second example: E8 = ΛC where C is the extended Hammingcode (8, 4).

Binary Construction ALet H8 be the extended Hamming code, with generator matrix

1 0 0 1 0 1 1 00 1 0 1 0 1 0 10 0 1 1 0 0 1 10 0 0 0 1 1 1 1

.

Since H8 is doubly even and self-dual, (1/√

2)ΛC is an evenunimodular lattice. Its generator matrix can be easily found:

1 0 0 0 1 1 1 00 1 0 0 1 1 0 10 0 1 0 1 0 1 10 0 0 1 0 1 1 10 0 0 0 2 0 0 00 0 0 0 0 2 0 00 0 0 0 0 0 2 00 0 0 0 0 0 0 2

.

Construction B

Let C be a code such that all codewords have even weight. TheConstruction B of C is defined as:

Λ(B)C =

{xxx ∈ Zn : xxx mod q ∈ C and

∑xi is divisible by 4

}.

Clearly Λ(B)C ⊂ ΛC . We have

λ(Λ(B)C )2 = min {8, dHam(C )} .

and |C | = 2n−1/ det Λ.

E8 revisited

Applying construction B to the very simple code

C = {(00000000), (11111111)} ∈ Z22, we get Λ

(B)C = E8.

Lattice Coding for Gaussian and Rayleigh Fading Channels

I Gaussian Channel:I Important parameters: packing density, kissing number and

shaping of a finite subset.I Lattice codes achieve channel capacity C = 1

2 log(1 + P/σ2) -short course by R. Zamir.

I Rayleigh Fading Channel

I Here it is important to consider lattices with full diversity andbig minimum product distance, since the transmission errorprobability is limited by a factor inversely proportional to theseparameters

I Algebraic Constructions: short course by J.-C. Belfiore andF. Oggier

Algebraic Constructions

Applications: Lattice Coding for Rayleigh Fading ChannelI Rayleigh Fading Channel

I Diversity of a lattice = minimum number k such that anynon-vanishing vector has at least k non-zero coordinates.

I A full rank lattice has full diversity if k = n.

I Minimum Product Distance of a full diversity lattice =Absolute value of minimum product of the coordinatesconsidering all (normalized) non-vanishing vectors of thislattice.

I This number is also very hard to obtain for general lattices,therefore the need for lattices constructed via AlgebraicNumber Theory

Lattice Coding for the Rayleigh Fading Channel

Product distance:

Diversity 1 Diversity 2. Product Distance: 0.25

Lattice Coding for the Rayleigh Fading Channel

I Rayleigh Fading ChannelI Lattices Constructed via Algebraic Number TheoryI K = Number field of degree n, F : K → Cn, Minkowski (or

twisted) homomorphism. If F is totally real, the image is alattice with full diversity. For this lattice we can determineboth packing density and minimum product distnace.

I Several papers on algebraic constructions of “rotated” versionsof special lattices (particularly Zn), and of the densest knownlattices.

I General Reference: E. Bayer-Flucker, Lattices and NumberFields.

Lattice Coding for Rayleigh Fading Channel

Good lattice constellations for both fading and Gaussian channels.

Lattices and Spherical Codes

I A spherical code is a finite subset of a sphere

Sn−1 = {xxx ∈ Zn : ‖xxx‖ = 1}

8-PSK

Spherical Codes

I Good spherical codes: greater minimum distance for fixednumber of points, easy coding and decoding process.

Lattices and Spherical Codes in Even Dimension

General reference: S. Costa, C. Torezzan, A. Campello and V.Vaishampayan, Flat Tori, Lattices, and Spherical Codes, ITA 2013Spherical Codes.

2πa

φ

φa : R → R2

φa(t) = a(cos(ta

), sin

(ta

))

I The lattice Λ =⟨

2πan

⟩is mapped onto a n-PSK (n = 8 here)

on a circle of radius a.

Flat Torus Map

a = (a1, a2) ∈ R2, a21 + a2

2 = 1

φa : R2 → R4

(u, v)→(a1 cos

u

a1, a1 sin

u

a1, a2 cos

u

a2, a2 sin

u

a2

)

φa is a double periodic map and φa(R2) = φa([0, 2πa1]× [0, 2πa2])(the parallel sides are identified).

Any lattice in R2 which contains 〈(2πa1, 0), (0, 2πa2)〉 induces aspherical code in R4 - a commutative group or a Slepian-type code.

Flat Torus Map

Flat Torus Map

Flat Torus Map

Flat Torus Map

Constructive Spherical Codes in R2n

I From good packing density lattices in Rn mapped on layers offlat tori.

I Easy construction, decoding in half of the dimensions

I [Torezzan, Costa, Vaishampayan ’13]

Discrete X Continuous Spherical Codes

Classical Spherical Codes

Given a certain distance d , what is the maximum number of pointsthat can be placed in a sphere so that their pairwise distances areat least d

Curve Packing Problem

Given a certain distance between laps (small-ball radius), what isthe longest curve that can be constructed satisfying the averagepower constraint

Continuous Spherical Codes

Continuous Spherical Codes

P1

P1

P2 P2

P3

P3

0

1

Curve Properties

I Average Power Constrain

I Must be twisted (or folded)

I To prevent large errors: distance between laps must be large

I In the low noise regime (small errors)

I Big length ⇒ Good Resolution

s

s(x1)r

s(x1)z1

y1

s(x2)

z2

y2

s(x2)

Curves on Flat Tori

I Building good curves can be achieved by building goodprojections [Vaishampayan and Costa ’03]

I Good curves (that outperform all previous ones) can beconstructed from projections of more general lattices[Campello, Torezzan and Costa ’13]

of the Zn lattice

I Any lattice can be approximated by projections of Zn [Sloane,Vaishampayan and Costa ’12]

I Any (n − k)- dimensional lattice can be approximated byprojections of any n-dimensional lattice. [Campello,Strapasson and Costa ’13].

More on Lattice Coding

You are strongly invited to attend the short courses:

“Lattice Codes for Signals and Newtorks” - R. Zamir,

“Explicit Lattice Constructions: From Codes to Number Fields” -J.-C. Belfiore and F. Oggier

Best wishes for your stay at the SPCoding School!

Special thanks to J. Strapasson, C. Torezzan, G. Jorge and C. Wink fordiscussions and suggestions.