Non-Binary Polar Codes using Reed-Solomon Codes and Algebraic - PowerPoint PPT Presentation

Non-Binary Polar Codes using Reed-Solomon Codes and Algebraic Geometry Codes Ryuhei Mori Toshiyuki Tanaka Graduate School of Informatics, Kyoto University Information Theory Workshop 2010

Contents Exponent of matrix ■ Reed-Solomon matrix - Our previous work ■ Simulation results - This work ■ Reed-Solomon matrix and Reed-Muller codes - This work ■ Hermitian codes - This work ■ 2 / 20

Exponent of matrix G : ℓ × ℓ matrix on F q . ■ P e ( G , n ): error probability of polar codes of length ℓ n = : N ■ (generator matrix is submatrix of G ⊗ n ). When rate of polar codes is smaller than capacity, for any ǫ > 0 N E ( G ) − ǫ ≤ − log P e ( G , n ) ≤ N E ( G ) + ǫ where E ( G ) ∈ [0, 1) is ℓ − 1 E ( G ) : = 1 � log ℓ D i ℓ i = 0 D i : partial distance [Korada, S ¸a¸ so˘ glu, and Urbanke 2009] [Arıkan and Telatar 2008] 3 / 20

Partial distance ℓ − 1 E ( G ) : = 1 � log ℓ D i ℓ i = 0 D i : partial distance D i : = d ( g i , � g i + 1 , ... , g ℓ − 1 � ) for 0 ≤ i ≤ ℓ − 2 D ℓ − 1 : = d ( g ℓ − 1 , 0) g i : i th row of G ■ � g i + 1 , ... , g ℓ − 1 � : a linear space spanned by g i + 1 , ... , g ℓ − 1 ■ d ( · , · ): Hamming distance ■     D 0 = 1 1 0 0 D 0 = 1 1 0 0  , D 1 = 1 1 0 1 D 1 = 2 1 0 1    D 2 = 3 1 1 1 D 2 = 2 1 1 0 D 0 D 1 D 2 = 3, D 0 D 1 D 2 = 4 4 / 20

Intuitive explanation D ( G , n ): a minimum distance of polar codes constructed from G ⊗ n P e ( G , n ) ≥ 2 − aD ( G , n ) for some constant a > 0 N E ( G ) − ǫ ≤ − log P e ( G , n ) ≤ N E ( G ) + ǫ N E ( G ) − ǫ ≤ D ( G , n ) ≤ N E ( G ) + ǫ where E ( G ) ∈ [0, 1) is ℓ − 1 E ( G ) : = 1 � log ℓ D i ℓ i = 0 5 / 20

Matrix transform   g 0 . .   .     g i − 1       g 0 g i + g j   .   G ′ = g i + 1 . ⇒ , for j > i G = =     .  .    .   . g ℓ − 1     g j   .   .   .   g ℓ − 1 The performance of SC decoder for polar codes is invariant under this transform Without loss of generality, we can assume D i = weight of i th row of G 6 / 20

Minimum distance of polar codes G : ℓ × ℓ matrix on F q ■ D i : weight of i th row of G ■ D i 1 , i 2 ,..., i n : weight of i th row of G ⊗ n where ℓ -ary expansion of i is i 1 ... i n ■ D i 1 , i 2 ,..., i n = D i 1 D i 2 · · · D i n From the law of large numbers, one has to choose an index i where number of a ∈ { 0, ... , ℓ − 1 } in i 1 ... i n is about n / ℓ Hence, one has to choose an index i such that � n � ℓ − 1 ℓ � D i 1 , i 2 ,..., i n ≈ D i i = 0 ℓ − 1 � � n 1 � = N E ( G ) = exp log D i ℓ i = 0 7 / 20

Contents Exponent of matrix ■ Reed-Solomon matrix - Our previous work ■ Simulation results - This work ■ Reed-Solomon matrix and Reed-Muller codes - This work ■ Hermitian codes - This work ■ 8 / 20

Matrix with large exponent If G doesn’t satisfy D 0 ≤ D 2 ≤ · · · ≤ D ℓ − 1 (1) there is a matrix G ′ which is obtained by permutation of rows of G such that E ( G ′ ) ≥ E ( G ) and G ′ satisfies (1) [Korada, S ¸a¸ so˘ glu, and Urbanke 2009] If (1) is satisfied, D i = minimum distance of � g i , ... , g ℓ − 1 � . Hence, obtaining large E ( G ) is equivalent to obtaining a sequence of linear codes C 1 , ... , C ℓ which satisfies C i : a linear code of dimension i and length ℓ ■ minimum distance of C i is large for i ∈ { 1, ... , ℓ } ■ C 1 ⊆ C 2 ⊆ · · · ⊆ C ℓ ■ Reed-Solomon codes have these properties. 9 / 20

Reed-Solomon matrix Let α be a primitive element of F q . A Reed-Solomon matrix G RS ( q ) is defined as α q − 2 α q − 3 · · · 1 0 α X q − 1  1 1 · · · 1 1 0  X q − 2 α ( q − 2)( q − 2) α ( q − 3)( q − 2) α q − 2 · · · 1 0   X q − 3  α ( q − 2)( q − 3) α ( q − 3)( q − 3) α q − 3  · · · 1 0   . .  . . . . .  . . . . . .   . . . · · · . . .     α q − 2 α q − 3 X · · · 1 0 α   1 1 1 · · · 1 1 1 Submatrix which consists of i th row to the last row is a generator matrix of extended Reed-Solomon code. The size ℓ of RS matrix is q . � � 1 0 Since G RS (2) = , RS matrix can be regarded as a generalization of Arıkan’s 1 1 � � 1 0 binary matrix . 1 1 Since D i = i + 1, E ( G RS ( q )) = log( q !) q log q 10 / 20

Exponent of Reed-Solomon matrix E ( G RS ( q )) = log( q !) q log q 2 4 16 64 256 q E ( G RS ( q )) 0.5 0.573120 0.691408 0.770821 0.822264 q →∞ E ( G RS ( q )) = 1 lim The exponent of binary matrix of size smaller than 32 is smaller than 0.55 [Korada, S ¸a¸ so˘ glu, and Urbanke 2009] Reed-Solomon matrix is useful for obtaining large exponent ! How about the performance for finite blocklength ? 11 / 20

Contents Exponent of matrix ■ Reed-Solomon matrix - Our previous work ■ Simulation results - This work ■ Reed-Solomon matrix and Reed-Muller codes - This work ■ Hermitian codes - This work ■ 12 / 20

Simulation P e ( W ( i ) � Error probability of polar codes ≤ N ) i ∈ F c � 1 � 0 Binary polar codes using vs 4-ary polar codes using G RS (4) 1 1 Same blocklength as binary codes 2 7 , 2 9 , 2 11 , and 2 13 AWGN( σ = 0.97865) Capacity is about 0.5 13 / 20

Simulation result 10 0 10 -1 Error probability 10 -2 N=2^7, 2^9, 2^11, 2^13 10 -3 binary polar codes 4-ary polar codes 10 -4 0.3 0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.5 Rate 14 / 20

Contents Exponent of matrix ■ Reed-Solomon matrix - Our previous work ■ Simulation results - This work ■ Reed-Solomon matrix and Reed-Muller codes - This work ■ Hermitian codes - This work ■ 15 / 20

Polar codes and Reed-Muller codes: binary case [Arıkan 2009] X : 1 0 ( X 2 , X 1 ) :(1, 1)(1, 0)(0, 1)(0, 0)  1 0 0 0  00 X 2 X 1 � 1 � X 0 X 2 1 1 0 0 01     1 1 1 X 1 1 0 1 0 10   1 1 1 1 1 11 { i ∈ { 0, ... , 2 n − 1 } | P e ( W ( i 1 ) ··· ( i n ) ) < ǫ } Polar rule: { i ∈ { 0, ... , 2 n − 1 } | i 1 + · · · + i n > k } Reed-Muller rule: � 1 � 0 Binary polar codes using and binary Reed-Muller codes are similar. 1 1 Reed-Muller rule maximizes the minimum distance. 16 / 20

Polar codes using RS matrix and Reed-Muller codes: q -ary case ( X 2 , X 1 ) : (2, 2) (2, 1) (2, 0) (1, 2) (1, 1) (1, 0) (0, 2) (0, 1) (0, 0) X 2 2 X 2   1 1 0 1 1 0 0 0 0 00 1 X 2 2 1 0 2 1 0 0 0 0 01 2 X 1   X 2   1 1 1 1 1 1 0 0 0 02 2   X 2 X 2   2 2 0 1 1 0 0 0 0 10 1     1 2 0 2 1 0 0 0 0 11 X 2 X 1     2 2 2 1 1 1 0 0 0 12 X 2   X 2   1 1 0 1 1 0 1 1 0 20   1   2 1 0 2 1 0 2 1 0 21 X 1   1 1 1 1 1 1 1 1 1 1 22 { i ∈ { 0, ... , q n − 1 } | P e ( W ( i 1 ) ··· ( i n ) ) < ǫ } Polar rule: { i ∈ { 0, ... , q n − 1 } | i 1 + · · · + i n > k } Reed-Muller rule: Q -ary polar codes using G RS ( q ) and q -ary Reed-Muller codes are also similar. { i ∈ { 0, ... , q n − 1 } | ( i 1 + 1) · · · ( i n + 1) > k } Hyperbolic rule: Hyperbolic rule maximizes the minimum distance (Massey-Costello-Justesen codes, hyperbolic cascaded RS codes). 17 / 20

Contents Exponent of matrix ■ Reed-Solomon matrix - Our previous work ■ Simulation results - This work ■ Reed-Solomon matrix and Reed-Muller codes - This work ■ Hermitian codes - This work ■ 18 / 20

Hermitian codes C i : a linear code of dimension i and length ℓ minimum distance of C i is large for i ∈ { 1, ... , ℓ } ■ C 1 ⊆ C 2 ⊆ · · · ⊆ C ℓ ■ Some class of algebraic geometry codes have the nested structure. G H ( q ): matrix using q -ary Hermitian codes q (even power of a prime) 4 16 64 256 E ( G RS ( q )) 0.573120 0.691408 0.770821 0.822264 E ( G H ( q )) 0.562161 0.707337 0.802760 0.859299 q 3 / 2 = size of G H ( q ) 8 64 512 4096 In order to obtain large exponent on fixed q , algebraic geometry codes are useful. 19 / 20

Conclusion Conclusion Reed-Solomon matrix has large exponent (previous work) ■ 4-ary polar codes using Reed-Solomon matrix has better performance than ■ � 1 � 0 binary polar codes using for finite blocklength 1 1 Polar codes using Reed-Solomon matrix, Reed-Muller codes, and ■ Massey-Costello-Justesen/hyperbolic cascaded RS codes are similar (generator matrices are constructed from G RS ( q ) ⊗ n ) Matrices using Hermitian codes have larger exponent than RS matrix ■ (unless q = 4). But size of the matrices are large. Future works Other heuristic decoding for q -ary polar codes using Reed-Solomon matrix ■ e.g., symbolwise/bitwise belief propagation. 20 / 20