Lattices from Codes or Codes from Lattices Amin Sakzad Dept of - PowerPoint PPT Presentation

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Lattices from Codes or Codes from Lattices Amin Sakzad Dept of Electrical and Computer Systems Engineering Monash University amin.sakzad@monash.edu Oct. 2013 Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Recall 1 Bounds Cycle-Free Codes and Lattices 2 Tanner Graph Lattices from Codes 3 Constructions Well-known high-dimensional lattices Codes from Lattices 4 Definitions Bounds Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Bounds Union Bound Estimate An estimate upper bound for the probability of error for a maximum-likelihood decoder of an n -dimensional lattice Λ over an unconstrained AWGN channel with noise variance σ 2 with coding gain γ (Λ) and volume-to-noise ratio α 2 (Λ , σ 2 ) : Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Bounds Union Bound Estimate An estimate upper bound for the probability of error for a maximum-likelihood decoder of an n -dimensional lattice Λ over an unconstrained AWGN channel with noise variance σ 2 with coding gain γ (Λ) and volume-to-noise ratio α 2 (Λ , σ 2 ) : �� πe P e (Λ , σ 2 ) � τ (Λ) � 4 γ (Λ) α 2 (Λ , σ 2 ) erfc , 2 where � ∞ 2 exp( − t 2 ) dt. √ π erfc(t) = t Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Bounds 0 10 −1 10 Normalizeed Error Probability (NEP) −2 10 Uncoded system −3 10 −4 10 Sphere bound −5 10 −6 10 −2 0 2 4 6 8 VNR(dB) Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Bounds Lower Bound on Probability of Error Theorem (Tarokh’99) If points of an n -dimensional lattice are transmitted over unconstrained AWGN channel with noise variance σ 2 , the probability of symbol error under maximum-likelihood decoding is lower-bounded as follows: � n � 1! + z 2 2 − 1 1 + z z P e (Λ , σ 2 ) ≥ e − z 2! + · · · + , � n � 2 − 1 where � n/ 2 � n z = α 2 (Λ , σ 2 )Γ 2 + 1 . Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Bounds Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Bounds Upper Bound on Coding Gain Theorem (Tarokh’99) Let ζ ( k ; P e ) denote the unique solution of equation (1 − erfc ( x )) 2 k = 1 − P e , and let n = 2 k , then: 1 γ (Λ) ≤ ζ ( k ; P e ) 2 ξ ( k ; P e ) . 4( k !) k , π where ξ ( k ; P e ) is the unique solution of x k − 1 � � 1 + x G k ( x ) � e − x 1! + · · · + = P e . ( k − 1)! Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Backgrounds Linear code C [ n, k, d min ] and its generator matrix G . Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Backgrounds Linear code C [ n, k, d min ] and its generator matrix G . Parity check matrix H . Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Backgrounds Linear code C [ n, k, d min ] and its generator matrix G . Parity check matrix H . Set r = n − k and rate is r = k n . Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Backgrounds Linear code C [ n, k, d min ] and its generator matrix G . Parity check matrix H . Set r = n − k and rate is r = k n . Message-Passing algorithms for decoding. Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Backgrounds Linear code C [ n, k, d min ] and its generator matrix G . Parity check matrix H . Set r = n − k and rate is r = k n . Message-Passing algorithms for decoding. Polynomial-time decoding algorithm if the corresponding “Tanner graph” has no cycle. Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Backgrounds Linear code C [ n, k, d min ] and its generator matrix G . Parity check matrix H . Set r = n − k and rate is r = k n . Message-Passing algorithms for decoding. Polynomial-time decoding algorithm if the corresponding “Tanner graph” has no cycle. Low-density Parity check (LDPC) code. Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Tanner graph constructions for codes Let H = ( h ij ) r × n be a parity check matrix for linear code C then we define Tanner graph of C as: Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Tanner graph constructions for codes Let H = ( h ij ) r × n be a parity check matrix for linear code C then we define Tanner graph of C as: Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Cycle free Tanner graphs Theorem (Etzion’99) Let C [ n, k, d min ] be a cycle free linear code of rate r ≥ 0 . 5 , then d min ≤ 2 . If r ≥ 0 . 5 , then � � � n + 1 � n < 2 d min ≤ + r . k + 1 k + 1 Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Tanner graph for lattices In the coordinate system S = { W i } n i =1 , a lattice Λ can be decomposed as Λ = Z n C (Λ) + L P (Λ) (1) where L ⊆ Z g 1 × Z g 2 × · · · × Z g n is the label code of Λ and C (Λ) = diag (det(Λ W 1 ) , . . . , det(Λ W n )) , P (Λ) = diag (det( P W 1 (Λ)) , . . . , det( P W n (Λ))) . Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Tanner graph for lattices In the coordinate system S = { W i } n i =1 , a lattice Λ can be decomposed as Λ = Z n C (Λ) + L P (Λ) (1) where L ⊆ Z g 1 × Z g 2 × · · · × Z g n is the label code of Λ and C (Λ) = diag (det(Λ W 1 ) , . . . , det(Λ W n )) , P (Λ) = diag (det( P W 1 (Λ)) , . . . , det( P W n (Λ))) . Tanner graph of a lattice Λ is the Tanner graph of its corresponding label code L . Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Tanner Graph Cycle-free lattices Theorem (Sakzad’11) Let Λ be an n -dimensional cycle-free lattice whose label code has rate greater than 0 . 5 . Then for a large even number n , the coding gain of Λ is γ (Λ) ≤ 2 n π . Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Constructions Backgrounds Construction A: Let C ⊆ F n 2 be a linear code. Define Λ as a lattice derived from C by: Λ = 2 Z n + C . Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Constructions Backgrounds Construction A: Let C ⊆ F n 2 be a linear code. Define Λ as a lattice derived from C by: Λ = 2 Z n + C . Construction D: Let C 0 ⊇ C 1 ⊇ · · · ⊇ C a be a family of a + 1 linear codes where C ℓ [ n, k ℓ , d ℓ min ] for 1 ≤ ℓ ≤ a and C 0 [ n, n, 1] 2 . Define Λ ⊆ R n as all vectors of the form trivial code F n k ℓ a c j β ( ℓ ) � � z + 2 ℓ − 1 , j ℓ =1 j =1 where z ∈ 2 Z n and β ( ℓ ) = 0 or 1 . j Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad

Recall Cycle-Free Codes and Lattices Lattices from Codes Codes from Lattices Constructions Minimum distance and coding gain Theorem (Barnes) Let Λ be a lattice constructed based on Construction D. Then we have �   d ℓ  min  d min (Λ) = min  2 , 2 ℓ − 1 1 ≤ ℓ ≤ a  where d ℓ min is the minimum distance of C ℓ for 1 ≤ ℓ ≤ a . Its coding gain satisfies kℓ � a n . γ (Λ) ≥ 4 ℓ =1 Lattice Coding II: Lattices from Codes or Codes from Lattices Amin Sakzad