List Decoding of Universal Polar Codes Boaz Shuval and Ido Tal 1/15

Overview � Universal polarization: � memoryless setting: ¸o˘ S ¸as glu & Wang, ISIT 2011 � settings with memory: Shuval & Tal, ISIT 2019 � Error probability analysis based on successive cancellation (SC) decoding � Potential for better results using successive cancellation list (SCL) decoding This talk: An efficient implementation of SCL for universal polarization 2/15

Setting � Communication with uncertainty: � Encoder: Knows channel belongs to a set of channels � Decoder: Knows channel statistics (e.g., via estimation) � Memory: � In channels � In input distribution � Universal polar code achieves: � Vanishing error probability over set � Best rate (infimal information rate over set) 3/15

List decoding improves probability of error! BSC 10 0 10 − 1 SC Word Error Rate upper 10 − 2 bound 10 − 3 10 − 4 10 − 5 0 . 25 0 . 3 0 . 35 0 . 4 0 . 45 0 . 5 Capacity 4/15

List decoding improves probability of error! BSC 10 0 L = 1 10 − 1 SC Word Error Rate upper 10 − 2 bound 10 − 3 10 − 4 10 − 5 0 . 25 0 . 3 0 . 35 0 . 4 0 . 45 0 . 5 Capacity 4/15

List decoding improves probability of error! BSC 10 0 L = 1 L = 2 L = 4 10 − 1 SC L = 8 Word Error Rate upper L = 16 10 − 2 bound 10 − 3 10 − 4 10 − 5 0 . 25 0 . 3 0 . 35 0 . 4 0 . 45 0 . 5 Capacity 4/15

List decoding improves probability of error! BEC 10 0 L = 1 L = 2 L = 4 10 − 1 SC L = 8 Word Error Rate upper L = 16 10 − 2 bound 10 − 3 10 − 4 10 − 5 0 . 25 0 . 3 0 . 35 0 . 4 0 . 45 0 . 5 Capacity 4/15

List decoding improves probability of error! AWGN 10 0 L = 1 L = 2 L = 4 10 − 1 SC L = 8 Word Error Rate upper L = 16 10 − 2 bound 10 − 3 10 − 4 10 − 5 0 . 25 0 . 3 0 . 35 0 . 4 0 . 45 0 . 5 Capacity 4/15

SCL in a nutshell � Parameter: list size L � Successively decode u 0 , u 1 , . . . , u N − 1 � Decoding path: sequence of decoding decisions � ˆ � ˆ u i − 1 = ˆ ˆ . . . u 0 u 1 u i − 1 � If u i is not frozen, split each decoding path into two paths, � ˆ � 0 u i − 1 ˆ u i − 1 � ˆ � 1 u i − 1 � For each i , keep L best decoding paths � Final decoding: choose best path 5/15

The Universal Construction � Concatenation of two transform blocks X N − 1 0 U N − 1 Y N − 1 fast slow channel 0 0 universal � Slow — for universality � Fast (Arıkan) — for vanishing error probability 6/15

The Universal Construction � Concatenation of two transform blocks slow fast slow X N − 1 fast 0 U N − 1 Y N − 1 slow channel 0 0 fast slow universal � Slow — for universality � Fast (Arıkan) — for vanishing error probability � Each block contains multiple copies of basic transforms � Polar: certain indices of U N − 1 are data bits, 0 other indices “frozen” (Honda & Yamamoto 2013) � Universal: data bit locations regardless of channel 6/15

Fast Transform vs. Slow Transform Fast (Arıkan) transform Slow transform 7/15

Fast Transform vs. Slow Transform Fast (Arıkan) transform Slow transform lateral lateral 7/15

Fast Transform vs. Slow Transform Fast (Arıkan) transform Slow transform lateral lateral medial lateral lateral 7/15

Fast Transform vs. Slow Transform Fast (Arıkan) transform Slow transform lateral i + 1 i lateral medial i i lateral lateral 7/15

The Rules of Slow lateral lateral � Any medial on left is a lateral medial − child of medial − and medial + lateral medial + on right medial − medial − medial + medial + medial − medial − medial + medial + lateral medial − medial + lateral medial − medial − medial + medial + medial − medial − medial + medial + lateral medial − lateral medial + lateral lateral 8/15

The Rules of Slow lateral lateral � Any medial on left is a lateral medial − child of medial − and medial + lateral medial + on right medial − medial − medial + medial + � Update of a medial + medial − medial − on left updates both medial + medial + medial − and medial + lateral medial − on right medial + lateral medial − medial − medial + medial + medial − medial − medial + medial + lateral medial − lateral medial + lateral lateral 8/15

The Rules of Slow lateral lateral � Any medial on left is a lateral medial − child of medial − and medial + lateral medial + on right medial − medial − medial + medial + � Update of a medial + medial − medial − on left updates both medial + medial + medial − and medial + lateral medial − on right medial + lateral medial − medial − Corollary medial + medial + Update of a medial + on left medial − medial − medial + medial + results in a cascade of lateral medial − updates, all the way to the lateral medial + channel on the far right lateral lateral 8/15

List Decoding — Fast Transform vs. Slow Transform Fast Transform Slow Transform The layer at depth λ gets The layer at depth λ gets updated once every 2 λ writes updated every other write to to the U vector the U vector Deep layers get updated All layers get updated infrequently frequently 9/15

List Decoding — Fast Transform vs. Slow Transform Fast Transform Slow Transform The layer at depth λ gets The layer at depth λ gets updated once every 2 λ writes updated every other write to to the U vector the U vector Deep layers get updated All layers get updated infrequently frequently Good news for sharing data Bad news for sharing data between list decoding paths between list decoding paths 9/15

Is All Hope Lost? Fast Transform Slow Transform � Layer at depth λ is � Layer at depth λ is updated once every 2 λ updated at every other writes write � At layer of depth λ , 2 λ � At layer of depth λ , at indices are updated at most two indices are once updated 10/15

Is All Hope Lost? Fast Transform Slow Transform � Layer at depth λ is � Layer at depth λ is updated once every 2 λ updated at every other writes write � At layer of depth λ , 2 λ � At layer of depth λ , at indices are updated at most two indices are once updated Our Saving Grace The updates are cyclic! 10/15

A Bespoke Data Structure � Aim: mimic the fast transform’s update regime � Frequent updates small � Infrequent large updates � Bespoke data structure: Cyclic Exponential Array � Main idea: an array comprised of sub-arrays, growing exponentially in length 11/15

Example entire array: cyclic exponential array: last written value 12/15

Example entire array: 0 cyclic exponential array: 0 last written value 12/15

Example entire array: 0 1 cyclic exponential array: 0 1 last written value 12/15

Example entire array: 0 1 2 cyclic exponential array: 0 1 2 last written value 12/15

Example entire array: 0 1 2 3 cyclic exponential array: 0 1 2 3 last written value 12/15

Example entire array: 0 1 2 3 4 cyclic exponential array: 0 1 2 3 4 last written value 12/15

Example entire array: 0 1 2 3 4 4 5 cyclic exponential array: 0 1 2 3 4 4 5 last written value 12/15

List Decoding of Universal Polar Codes Boaz Shuval and Ido Tal 1/15 - PowerPoint PPT Presentation

List Decoding of Universal Polar Codes Boaz Shuval and Ido Tal 1/15 Overview Universal polarization: memoryless setting: o S as glu & Wang, ISIT 2011 settings with memory: Shuval & Tal, ISIT 2019 Error

List Decoding for Universal Polar Codes Boaz Shuval and Ido Tal 1/15 Overview Universal

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