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Recursive Trellis Decoding Techniques of Polar Codes Peter Trifonov ITMO University, Russia June 18, 2020 Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 1 / 18

Outline Polar codes 1 Recursive trellis decoding 2 Application of recursive trellis decoding to polar codes 3 Hybrid decoding 4 Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 2 / 18

Polar codes Polar Codes � 1 � ⊗ m 0 Encoding: c n − 1 = u n − 1 B m , u i = 0 , i ∈ F 0 0 1 1 B m is the bit-reversal permutation matrix (it will be essential!) n = 2 m Successive cancellation decoding Far from maximum likelihood Decoding latency O ( n ) Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 3 / 18

Recursive trellis decoding Recursive Trellis Decoding: the Idea 10011100 Partition the noisy vector into a number of sections 0000 1100 1001 1111 Find the most probable codeword subvectors for each section 00 00 10 00 Combine short codeword subvectors 10 01 11 11 into longer subvectors Do this recursively 1 noisy vector 1T. Fujiwara, H. Yamamoto, T. Kasami, and S. Lin, “A trellis-based recursive maximum-likelihood decoding algorithm for binary linear block codes,” IEEE Transactions On Information Theory, vol. 44, no. 2, March 1998. Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 4 / 18

Recursive trellis decoding Sectionalized Trellis of a Linear Block Code Given a linear code C , let C h , h ′ be its subcode, such that all its codewords have non-zero symbols only in positions h ≤ i < h ′ Let p h , h ′ ( C ) be a linear code obtained by   puncturing all symbols, except those in positions 1 1 1 1 1 1 1 1   1 1 1 1 0 0 0 0 h ≤ i < h ′ , from codewords of C   C : G =   1 1 0 0 1 1 0 0 Let s h , h ′ ( C ) = p h , h ′ ( C h , h ′ ) , i.e. a code obtained 1 0 1 0 1 0 1 0 from C by shortening it on all symbols except those with indices h ≤ i < h ′ . Trellis paths from time h to time h ′ correspond to cosets in p h , h ′ ( C ) / s h , h ′ ( C ) . The same coset may occur several times in a trellis Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 5 / 18

Recursive trellis decoding Sectionalized Trellis of a Linear Block Code Given a linear code C , let C h , h ′ be its subcode, such that all its codewords have non-zero symbols only in positions h ≤ i < h ′   1 1 1 1 1 1 1 1 Let p h , h ′ ( C ) be a linear code obtained by   1 1 1 1 0 0 0 0   C : G = puncturing all symbols, except those in positions   1 1 0 0 1 1 0 0 h ≤ i < h ′ , from codewords of C 1 0 1 0 1 0 1 0   Let s h , h ′ ( C ) = p h , h ′ ( C h , h ′ ) , i.e. a code obtained 1 1 1 1 from C by shortening it on all symbols except   p 0 , 4 ( C ) : G 0 , 4 = 1 1 0 0 those with indices h ≤ i < h ′ . 1 0 1 0 Trellis paths from time h to time h ′ correspond to cosets in p h , h ′ ( C ) / s h , h ′ ( C ) . The same coset may occur several times in a trellis Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 5 / 18

Recursive trellis decoding Sectionalized Trellis of a Linear Block Code Given a linear code C , let C h , h ′ be its subcode,   such that all its codewords have non-zero 1 1 1 1 1 1 1 1 symbols only in positions h ≤ i < h ′   1 1 1 1 0 0 0 0   C : G =   Let p h , h ′ ( C ) be a linear code obtained by 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 puncturing all symbols, except those in positions   h ≤ i < h ′ , from codewords of C 1 1 1 1   Let s h , h ′ ( C ) = p h , h ′ ( C h , h ′ ) , i.e. a code obtained p 0 , 4 ( C ) : G 0 , 4 = 1 1 0 0 1 0 1 0 from C by shortening it on all symbols except those with indices h ≤ i < h ′ . � 1 1 1 1 � s 0 , 4 ( C ) : G 0 , 4 = Trellis paths from time h to time h ′ correspond to cosets in p h , h ′ ( C ) / s h , h ′ ( C ) . The same coset may occur several times in a trellis Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 5 / 18

Recursive trellis decoding Sectionalized Trellis of a Linear Block Code   1 1 1 1 1 1 1 1   1 1 1 1 0 0 0 0   Given a linear code C , let C h , h ′ be its subcode, C : G =   1 1 0 0 1 1 0 0 such that all its codewords have non-zero 1 0 1 0 1 0 1 0 symbols only in positions h ≤ i < h ′   1 1 1 1 Let p h , h ′ ( C ) be a linear code obtained by   p 0 , 4 ( C ) : G 0 , 4 = 1 1 0 0 puncturing all symbols, except those in positions 1 0 1 0 h ≤ i < h ′ , from codewords of C � 1 1 1 1 � s 0 , 4 ( C ) : G 0 , 4 = Let s h , h ′ ( C ) = p h , h ′ ( C h , h ′ ) , i.e. a code obtained from C by shortening it on all symbols except 0000 those with indices h ≤ i < h ′ . 0000 0000 1111 1111 Trellis paths from time h to time h ′ correspond to 0011 1100 0011 1100 1100 s e cosets in p h , h ′ ( C ) / s h , h ′ ( C ) . The same coset may 1010 0101 0101 1010 1010 0110 0110 occur several times in a trellis 1001 1001 0110 Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 5 / 18

Recursive trellis decoding Recursive Trellis Decoding of Linear Block Codes For each coset D ∈ p x , y ( C ) / s x , y ( C ) find the most probable element l ( D ) , and its correlation m ( D ) Composite branch table CBT x , y stores ( l ( D ) , m ( D )) ML decoding of ( n , k ) code C : p 0 , n ( C ) / s 0 , n ( C ) contains a single element, so CBT 0 , n contains the solution of the ML decoding problem Construction of CBT x , y : y − x ≥ 2: Consider all combinations of cosets D ′ ∈ p x , z ( C ) / s x , z ( C ) , 0000 0000 0000 D ′′ ∈ p z , y ( C ) / s z , y ( C ) , such that 1111 1111 0011 1100 0011 D ′ . D ′′ = D ∈ p x , y ( C ) / s x , y ( C ) , i.e. concatenation of any 1100 1100 s e 1010 0101 their representatives is in D ∈ p x , y ( C ) / s x , y ( C ) 0101 1010 1010 0110 0110 m ( D ) = max D ′ , D ′′ ( m ( D ′ ) + m ( D ′′ )) , l ( D ) = l ( D ′ ) . l ( D ′′ ) 1001 1001 0110 Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 6 / 18

Recursive trellis decoding Generator Matrices of Section Codes   G ( s ) 0 x , z   G ( s ) 0   Generator matrix of p x , y ( C ) is G ( p ) z , y x , y =   G ( 00 ) G ( 01 )   x , y x , y G ( 10 ) G ( 11 ) x , y x , y   G ( s ) 0 x , z   Generator matrix of s x , y ( C ) is G ( s ) G ( s ) x , y =  0  z , y G ( 00 ) G ( 01 ) x , y x , y G ( 00 ) x , y , G ( 01 ) x , y are some k ′′ x , y × ( z − x ) and k ′′ x , y × ( y − z ) matrices G ( 10 ) x , y , G ( 11 ) x , y are some k ′ x , y × ( z − x ) and k ′ x , y × ( y − z ) matrices � � G ( 10 ) G ( 11 ) One-to-one correspondence between vG ′ x , y , where G ′ x , y = , and x , y x , y cosets D ∈ p x , y ( C ) / s x , y ( C ) k ′ x , y CBT entries are indexed by v ∈ F 2 Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 7 / 18

Recursive trellis decoding Merging the Composite Branch Tables k ′ CBT x , y [ v ] . m = max ( CBT x , z [ a ] . m + CBT z , y [ b ] . m ) , v ∈ F x , y 2 k ′′ x , y u ∈ F 2 where a and b are indices of the cosets D ′ ∈ p x , z ( C ) / s x , z ( C ) and D ′′ ∈ p z , y ( C ) / s z , y ( C ) , such � � � � G ( 00 ) G ( 01 ) � � � � ∈ D ′ and x , y x , y that u v u v ∈ D ′′ G ( 10 ) G ( 11 ) x , y x , y Such values a , b can be identified from � � � � a � � � b � � � G ( 00 ) G ( 01 ) � a ′ G ( s ) � u v � � b ′ G ( s ) � u v � x , y x , y x , z z , y = = , G ( 10 ) G ( 11 ) G ′ G ′ x , z x , y z , y x , y where a ′ , b ′ are some irrelevant values � u v � � � u v � � G x , y for some � G x , y and � The solutions are a = G x , y and b = G x , y Complexity is O ( 2 k ′ x , y + k ′′ x , y ) Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 8 / 18

Application of recursive trellis decoding to polar codes Recursive Trellis Decoding of Polar Codes Arikan successive cancellation decoding is equivalent 2 to successive application of the recursive trellis decoding algorithm to codes generated by rows i , . . . , 2 m − 1 of A m Recursive trellis algorithm provides ML decoding, SC algorithm is far from ML Can polar codes be efficiently decoded by a single pass of the recursive trellis algorithm? 2P . Trifonov. Trellis-based Decoding Techniques for Polar Codes with Large Kernels. In Proc. of IEEE Information Theory Workshop 2019 Peter Trifonov ( ITMO University, Russia) Recursive Trellis Decoding Techniques of Polar Codes June 18, 2020 9 / 18