Lattices and Lattice Codes

Transcription

1 Lattices and Lattice Codes Trivandrum School on Communication, Coding & Networking January 27 30, 2017 Lakshmi Prasad Natarajan Dept. of Electrical Engineering Indian Institute of Technology Hyderabad 1 / 32

2 Recall Communications I & II 2 / 32

3 Capacity - Fundamental limit for Digital Communications C(SNR) = log 2 (1 + SNR): Max. spectral efficiency at a given SNR for reliable communication. SNR (η) = 2 η 1: Min. SNR required for a given η for reliable communication. A code C with spectral efficiency η is good if it has negligible P e for SNR > 2 η 1. 3 / 32

4 Dr. Lalitha Vadlamani s talk: Coding Theory I & II 4 / 32

5 How to Design Good Communication Schemes? 1 Combining Long Binary Codes with Small Modulation Schemes Examples: Bit-interleaved Coded Modulation, Multilevel Codes 2 Directly map messages to points in high-dimensional space R N Examples: Lattice codes, Group codes 5 / 32

6 This talk: Directly map messages to vectors ˆ Code (over R) C = {s 1,..., s M } R N ˆ Power P = 1 N s s M 2 M ˆ Noise variance σ 2 = N o (per dimension) 2 ˆ Signal to noise ratio SNR = P σ 2 = 2P N o ˆ Spectral Efficiency η = 2 log 2 M bits/s/hz (assuming N = 2T W ) N ˆ Probability of Error P e = P ( ˆm m) 6 / 32

7 References ˆ F. Oggier and E. Viterbo, Algebraic Number Theory and Code Design for Rayleigh Fading Channels. Foundations and Trends in Communications and Information Theory, vol. 01, no. 03, Section 3 of this reference is a primer on lattices. ˆ J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups. New York, NY, USA: Springer-Verlag, ˆ R. Zamir, Lattice Coding for Signals and Networks. Cambridge, UK: Cambridge University Press, ˆ G. D. Forney, Multidimensional constellations. II. Voronoi constellations, IEEE Journal on Selected Areas in Communications, vol. 7, no. 6, pp , Aug ˆ H. A. Loeliger, Averaging bounds for lattices and linear codes, IEEE Transactions on Information Theory, vol. 43, no. 6, pp , Nov ˆ M. P. Wilson, K. Narayanan, H. D. Pfister and A. Sprintson, Joint Physical Layer Coding and Network Coding for Bidirectional Relaying, IEEE Transactions on Information Theory, vol. 56, no. 11, pp , Nov / 32

8 1 Lattices 2 Sphere Packing Problem and Coding for Gaussian Channel 3 Construction A: From Error Correcting Codes to Lattices 4 Nested Lattice Codes/Voronoi Constellations 8 / 32

9 Lattices Say g 1,..., g m R N are linearly independent column vectors, m N. Definition The lattice Λ generated by g 1,..., g m is the set of all integer linear combinations of g 1,..., g m Λ = {z 1 g z m g m z i Z} ˆ We will consider only full-rank lattices m = N ˆ The matrix G = [g 1 g 2 g N ] is a generator matrix for Λ Λ = { Gz z Z N } G R N N is square, rank(g) = N, i.e., det(g) 0. ˆ {g 1,..., g N } forms a basis for Λ. ˆ Generator matrix G of a lattice Λ is not unique. 8 / 32

10 Examples of lattices in two dimensions 9 / 32

11 A Lattice of Flowers Trivandrum Flower Show, Kanakakunnu Palace Grounds 28 Jan / 32

12 Algebraic Properties of Lattices Let Λ R N be any N-dimensional lattice. ˆ The all-zero vector 0 is a lattice point 0 = 0 g g N Λ ˆ Additive inverse of any lattice point is a lattice point λ = z 1 g z N g N Λ λ = z 1 g 1 z N g N Λ ˆ Sum of any two lattice points is also a lattice point λ z = z 1 g z N g N and λ u = u 1 g u N g N λ z + λ u = (z 1 + u 1 )g (z N + u N )g N Λ ˆ If a Z and λ Λ then aλ Λ. Properties similar to vector space, but Z is NOT a field!! Every lattice Λ is a module over Z 11 / 32

13 Minimum Distance of a Lattice Minimum Distance d min (Λ) Smallest distance between any two lattice points d min = min λ 1 λ 2 λ 1 λ 2 Note that λ 1 λ 2 Λ and λ 1 λ 2 0. Hence, d min = min λ = min norm of non-zero lattice points λ 0 Note Λ is invariant to translation by any lattice vector λ Λ, Λ = λ + Λ ˆ The distance of any nearest neighbor for any lattice point is d min 12 / 32

14 Examples in two dimensions 13 / 32

15 Voronoi Region V(Λ). 0 ˆ Closest vector lattice quantizer Q Λ (x) = λ if λ is the nearest lattice point to x R N ˆ Fundamental Voronoi region V = Q 1 (0) set of all points closer to 0 than any other lattice point Λ ˆ λ + V = Q 1 Λ (λ) all points mapped to λ by Q Λ ˆ Translates {λ + V} of V are non-intersecting and cover R N 14 / 32

16 Volume and Hermite Parameter Volume of the lattice V (Λ) V (Λ) = volume ( V(Λ) ) is the volume of the Voronoi region ˆ If G is a generator matrix, then V (Λ) = det(g) ˆ This is a lattice invariant, does not depend on the choice of G How to measure the density of lattice points ˆ Number of lattice points per unit volume in R N = 1/V (Λ) ˆ By simply scaling Λ aλ, we can trivially modify the number of lattice points per unit volume ˆ We need a measure that is invariant to scaling. Hermite Parameter γ(λ) γ(λ) = d2 min (Λ) V (Λ) 2/N ˆ γ(λ) N/2 = density when the lattice scaled to unit min distance. ˆ γ(λ) = squared min dist when the lattice is scaled to unit volume. 15 / 32

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18 1 Lattices 2 Sphere Packing Problem and Coding for Gaussian Channel 3 Construction A: From Error Correcting Codes to Lattices 4 Nested Lattice Codes/Voronoi Constellations 17 / 32

19 The Sphere Packing Problem A sphere packing is an arrangement of infinitely-many non-overlapping identical spheres in the Euclidean space. ˆ A lattice can be used to pack spheres: place a sphere centered at each lattice point ˆ Spheres must be non-overlapping: largest spheres that can be packed using Λ have radius r pack = d min 2 This is called the packing radius of Λ Sphere packing problem What is the maximum density (number of spheres per unit volume) with which spheres can be packed in R N? 17 / 32

20 Sphere Packing and Hermite Parameter Center Density δ(λ) = rn pack (Λ) V (Λ) density of spheres when the lattice is scaled to unit packing radius ( γ(λ) Using d min = 2r pack, we obtain δ(λ) = 4 ) N/2 18 / 32

21 Best Lattice Sphere Packings Let γ N = largest Hermite parameter among all lattices in R N Theorem For all sufficiently large N, we have N 2πe γ N N 2πe ˆ Bounds on packing efficiency increase with the dimension N. Best Lattice Packings for N 8 N Λ Z A 2 D 3 D 4 D 5 E 6 E 7 E 8 γ N κ / 32

22 From Lattices to Codes ˆ A code C = {s 1,..., s M } for the (vector) Gaussian channel is a finite set of points in R N. ˆ Replace the average power constraint with the more stringent per-codeword power constraint: s i 2 N P s i NP ˆ Let B be the N-dimensional ball of radius NP centered at 0 s i B for i = 1,..., M Spherically-shaped lattice codes We can carve an N-dimensional code C from a lattice Λ by setting C = Λ B 20 / 32

23 Spherically-Shaped Lattice Codes Say, we desire to construct a code for given N, P & spectral efficiency η M = C Vol(B) V (Λ) V (Λ) Vol(B) 2 Nη/2 Then d 2 min = γ(λ) V (Λ)2/N γ(λ) Vol(B) 2/N 2 η To construct a good code, we require Λ to have a large γ(λ). 21 / 32

24 1 Lattices 2 Sphere Packing Problem and Coding for Gaussian Channel 3 Construction A: From Error Correcting Codes to Lattices 4 Nested Lattice Codes/Voronoi Constellations 22 / 32

25 Prime Fields F p Prime fields are finite fields whose cardinality is a prime number. Let p 2 be any prime integer. Definition The prime field of cardinality p is the set F p = {0, 1,..., p 1} where addition and multiplication are performed modulo p ˆ a b = (a + b) mod p, and ˆ a b = (a b) mod p. ˆ F p satisfies all the axioms of a field: existence of additive & multiplicative identities and inverses, distributive law, etc. Addition in F 3 = {0, 1, 2} Multiplication in F 3 = {0, 1, 2} / 32

26 Linear Codes over F p Addition of vectors over F p : a b = (a 1,..., a N ) (b 1,..., b N ) = (a 1 b 1,..., a N b N ) Definition A linear code C over F p of length N is a subspace of F N p. ˆ C = p k, where integer k is the dimension of the code. Example ˆ N = 5, p = 2 and C = {(0, 0, 0, 0, 0), (1, 1, 1, 1, 1)}. This is the repetition code of length 5 over F 2. ˆ N = 3, p = 3 and C = {(0, 0, 0), (1, 1, 1), (2, 2, 2)}. This is the repetition code of length 3 over F / 32

27 Lattices from Codes The linear code C F n p is closed under addition mod p c 1, c 2 C (c 1 + c 2 ) mod p C Example: p = 7, N = 2 C = {(0, 0), (3, 1), (6, 2), (2, 3), (5, 4), (1, 5), (4, 6)} ˆ Embed C into R N using natural map Create a lattice Λ by tiling copies of C in R N 24 / 32

28 Lattices from Codes: Construction A Λ = C + p Z N = u Z n (C + p u) Example (continued) [ ] 3 1 G = 1 2 p = 2: Using codes over F 2 = {0, 1}, C F N 2 p = 2 and, say, C = 2 k, d H = min Hamming distance Vol(Λ) = 2 (N k) and d min (Λ) = min{2, d H } 25 / 32

29 Examples of Construction A Lattices from Binary Codes 1 The D 4 lattice Let C = {(0, 0, 0, 0), (1, 1, 1, 1)} be repetition code of length N = 4. Dimension k = 1 and min Hamming distance d H = 4. 2 The E 8 lattice V (Λ) = 8 and d min = 2 γ(λ) = G = Use C = [N = 8, k = 4, d H = 4] extended Hamming code. V (Λ) = 16 and d min = 2 γ(λ) = 2 D 4 and E 8 are the densest lattices in their dimensions. Theorem For N large, there exists a Construction A lattice with γ N 2πe 26 / 32

30 1 Lattices 2 Sphere Packing Problem and Coding for Gaussian Channel 3 Construction A: From Error Correcting Codes to Lattices 4 Nested Lattice Codes/Voronoi Constellations 27 / 32

31 Modulo Lattice Operation x mod Λ = x Q Λ (x) R N V(Λ) Modulo operation lends algebraic structure to the Voronoi region V Λ V(Λ) V(Λ) (x, y) V(Λ) (x + y) mod Λ ˆ The origin 0 V(Λ), and for every x V, there exists a unique y = x mod V such that (x + y) mod Λ = 0 ˆ V(Λ) is a group under addition modulo Λ. 27 / 32

32 Nested Lattices and Lattice Codes/Voronoi Constellations Nested Lattices Lattice Codes or Voronoi Constellations ˆ Λ s Λ are lattices ˆ Λ s is a subgroup of Λ ˆ Λ/Λ s = Λ V Λs is a group Addition: x y = (x + y) mod Λ s 28 / 32

33 Lattice Codes Lattice Code Λ/Λ s ˆ Finite group under addition mod Λ s ˆ Λ/Λ s = V (Λ s )/V (Λ) ˆ η = 2 N log 2 V (Λ s) V (Λ) Coding lattice (Fine lattice) Λ ˆ Provides noise resilience ˆ Want large d min (Λ) & small V (Λ) Shaping lattice (Coarse lattice) Λ s ˆ Carves a finite code from Λ ˆ Determines transmit power ˆ Want small power & large V (Λ s ) Lattice codes are good for many things: achieve capacity in Gaussian channel and dirty paper channel, Diversity-Multiplexing Trade-off in multiple-antenna channels, relay networks (compute & forward), wiretap channels, interference channels, quantization, cryptography, etc. etc. etc. 29 / 32

34 Example Application: Bidirectional Relay Channel 30 / 32

35 Joint PHY/Network-layer Lattice Coding Scheme Wilson, Narayanan, Pfister and Sprintson, IEEE Trans. Inf. Theory, Nov / 32

36 Conclusion ˆ Lattices have rich algebraic and geometric properties. ˆ They have strong connection to coding/modulation for Gaussian channel, and to error correcting codes over finite fields. ˆ Structured codes can be obtained using nested lattice pairs. This structure can be useful in communication engineering, for instance for exploiting interference in wireless channels. ˆ Lattices have several other applications: cryptography, information-theoretic security, codes for fading and multiple antenna channels, vector quantization etc. Thank You!! 32 / 32