An Introduction to Lattices and their Applications in Communications Frank R. Kschischang Chen Feng University of Toronto, Canada 2014 Australian School of Information Theory University of South Australia Institute for Telecommunications Research Adelaide, Australia November 13, 2014 Outline 1 Fundamentals 2 Packing, Covering, Quantization, Modulation 3 Lattices and Linear Codes 4 Asymptopia 5 Communications Applications 2 Part 1: Lattice Fundamentals 3 Notation • C: the complex numbers (a field) • R: the real numbers (a field) • Z: the integers (a ring) • X n: the n-fold Cartesian product of set X with itself; n X = f(x1;:::; xn): x1 2 X ; x2 2 X ;:::; xn 2 X g. If X is a field, then the elements of X n are row vectors. • X m×n: the m × n matrices with entries from X . ∗ • If (G; +) is a group with identity 0, then G , G n f0g denotes the nonzero elements of G. 4 Euclidean Space Lattices are discrete subgroups (under vector addition) of finite-dimensional Euclidean spaces such as n. R x d( • x; y) In n we have R •y Pn (0; 1) k • an inner product: hx; yi xi yi x , i=1 k k p ky • a norm: kxk , hx; xi • 0 • a metric: d(x; y) , kx − yk (1; 0) • Vectors x and y are orthogonal if hx; yi = 0. n • A ball centered at the origin in R is the set n Br = fx 2 R : kxk ≤ rg: n • If R is any subset of R , the translation of R by x is, for any n x 2 R , the set x + R = fx + y : y 2 Rg. 5 Lattices Definition n Given m linearly independent (row) vectors g1;:::; gm 2 R , the lattice Λ generated by them is defined as the set of all integer linear combinations of the gi 's: ( m ) X Λ(g1;:::; gm) , ci gi : c1 2 Z; c2 2 Z;:::; cm 2 Z : i=1 • g1; g2;:::; gm: the generators of Λ • n: the dimension of Λ • m: the rank of Λ • We will focus only on full-rank lattices (m = n) in this tutorial 6 1 2 1 2 Example: Λ 2; 3 ; 2; −3 −3g2 3g1 + g2 g1 0 g2 7 3 2 Example: Λ 2; 3 ; (1; 0) + g2 3g1 g1 −3g2 0 g2 8 Generator Matrix Definition n A generator matrix GΛ for a lattice Λ ⊆ R is a matrix whose rows generate Λ: 2 3 g1 6 . 7 n×n n GΛ = 4 . 5 2 R and Λ = fcGΛ : c 2 Z g: gn Example: 1=2 2=3 3=2 2=3 G = and G = 1 1=2 −2=3 2 1 0 generate the previous examples. By definition, a generator matrix is full rank. 9 When do G and G0 Generate the Same Lattice? n×n Recall that a matrix U 2 Z is said to be unimodular if −1 n×n −1 det(U) 2 f1; −1g. If U is unimodular, then U 2 Z and U is also unimodular. (U is unimodular $ det(U) is a unit.) Theorem 0 n×n Two generator matrices G; G 2 R generate the same lattice if n×n and only if there exists a unimodular matrix U 2 Z such that G0 = UG. (In any commutative ring R, for any matrix A 2 Rn×n, we have A adj(A) = det(A)In, where adj(A), the adjugate of A is given by i+j [adj(A)]i;j = (−1) Mj;i where Mj;i is the minor of A obtained by deleting the jth row and ith column of A. Note that adj(A) 2 Rn×n. The matrix A is invertible (in Rn×n) if and only if det(A) is an invertible element (a unit) of R, in which case A−1 = (det(A))−1 adj(A). cf. Cramer's rule.) 10 Proof For \)": Assume that G and G0 generate the same lattice. Then there are integer matrices V and V0 such that G0 = VG and G = V0G0: Hence, G0 = VV0G0 = (VV0)G0; from which it follows that VV0 is the identity matrix. However, since det(V) and det(V0) are integers and the determinant function is multiplicative, we have det(V) det(V0) = 1. Thus det(V) is a unit in Z and so V is unimodular. For \(": Assume that G0 = UG for a unimodular matrix U, let Λ be generated by G and let Λ0 be generated by G0. An element 0 0 n λ 2 Λ can be written, for some c 2 Z as 0 0 0 0 n λ = cG = cUG = c G 2 Λ, which shows, since c = cU 2 Z , that Λ0 ⊆ Λ. On the other hand, we have G = U−1G0 and a similar argument shows that Λ ⊆ Λ0. 11 Lattice Determinant Definition The determinant, det(Λ), of a full-rank lattice Λ is given as det(Λ) = j det(GΛ)j where GΛ is any generator matrix for Λ. • Note that, in view of the previous theorem, this is an invariant of the lattice Λ, i.e., the determinant of Λ is independent of the choice of GΛ. • As we will now see, this invariant has a geometric significance. 12 Fundamental Region Definition n n A set R ⊆ R is called a fundamental region of a lattice Λ ⊆ R if the following conditions are satisfied: n S 1 R = λ2Λ(λ + R). 2 For every λ1; λ2 2 Λ with λ1 6= λ2,(λ1 + R) \ (λ2 + R) = ;. In other words, the translates of a fundamental region R by lattice n points form a disjoint covering (or tiling) of R . • A fundamental region R cannot contain two points x1 and x2 whose difference is a nonzero lattice point, since if x1 − x2 = λ 2 Λ, λ 6= 0, for x1; x2 2 R, we would have x1 2 0 + R and x1 = x2 + λ 2 λ + R, contradicting Property 2. • Algebraically, the points of a fundamental region form a complete system of coset representatives of the cosets of Λ in n R . 13 Fundamental Regions for Λ((1=2; 2=3); (1=2; −2=3)) • Each shaded fundamental region serves as a tile; the union of translates of a tile by all lattice points forms a disjoint covering 2 of R . • Fundamental regions need not be connected sets. 14 Fundamental Parallelepiped Definition The fundamental parallelepiped of a generating set n g1;:::; gn 2 R for a lattice Λ is the set ( n ) X n P(g1;:::; gn) , ai gi :(a1;:::; an) 2 [0; 1) : i=1 g1 g1 0 0 g2 g2 1 2 1 2 3 2 P 2 ; 3 ; 2 ; − 3 P 2 ; 3 ; (1; 0) 15 Their Volume = det(Λ) Proposition Given a lattice Λ, the fundamental parallelepiped of every generating set for Λ has the same volume, namely det(Λ). Proof: Let g1;:::; gn form the rows of a generator matrix G. Then, by change of variables, n Vol(P(g1;:::; gn)) = Vol(faG : a 2 [0; 1) g) = Vol([0; 1)n) · j det(G)j = j det(G)j = det(Λ) 16 All Fundamental Regions Have the Same Volume Proposition More generally, every fundamental region R of Λ has the same volume, namely det(Λ). Proof (by picture): Proof (by mapping): translate each point of R by some lattice vector to a unique point of P. Partition R into \pieces" R1; R2;::: translated by the same vector. If the pieces each have a well-defined volume, then P Vol(R) = i Vol(Ri ), and the result follows since volume is translation invariant and the union of the translated pieces is P. 17 Voronoi Region Definition n Given a lattice Λ ⊆ R and a point λ 2 Λ, a Voronoi region of λ is defined as n 0 0 0 V(λ) = fx 2 R : 8λ 2 Λ; λ 6= λ; kx − λk ≤ kx − λ kg; where ties are broken systematically. The Voronoi region of 0 is often called the Voronoi region of the lattice and it is denoted by V(Λ). 18 Nearest-Neighbor Quantizer Definition (NN) n A nearest neighbor quantizer QΛ : R ! Λ associated with a lattice Λ maps a vector to the closest lattice point Q(NN)(x) = arg min kx − λk; Λ λ2Λ where ties are broken systematically. • The inverse image (NN) −1 [QΛ ] (λ) is a Voronoi region of λ. • QΛ(x) may be difficult to n compute for arbitrary x 2 R . 19 Minimum Distance Definition n The minimum distance of a lattice Λ ⊆ R is defined as dmin(Λ) = min kλk: λ2Λ∗ Fact: (Λ) d min dmin(Λ) > 0 Proof: exercise. 20 Successive Minima Recall that Br denotes the n-dimensional ball of radius r centered at n the origin: Br , fx 2 R : kxk ≤ rg. Definition n For a lattice Λ ⊂ R , let Li (Λ) , minfr : Br contains at least i linearly indep. lattice vectorsg: Then L1 ≤ L2 ≤ ::: ≤ Ln are the successive minima of Λ. • We have L1(Λ) = dmin(Λ). Here L2 > L1 • Note that Ln(Λ) contains n linearly independent lattice vectors by definition, but these may not generate Λ! (Example: 5 5 2Z [ (1; 1; 1; 1; 1) + 2Z has L1 = ··· = L5 = 2, but the 5 linearly independent vectors in B2 generate only 5 2Z .) 21 A Quick Recap n As subgroups of R , lattices have both algebraic and geometric properties. • Algebra: closed under subtraction (forms a subgroup) • Geometry: fundamental regions (fundamental parallelepiped, Voronoi region), (positive) minimum distance, successive minima • Because lattices have positive minimum distance, they are n discrete subgroups of R , i.e., surrounding the origin is an open ball containing just one lattice point (the origin itself).