Non-Uniform Windowed Decoding For Multi-Dimensional Spatially-Coupled LDPC Codes Lev Tauz, Homa Esfahanizadeh, and Lara Dolecek Electrical and Computer Engineering University of California, Los Angeles ISIT 2020 Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 1 / 34
Outline Background and Previous Work 1 Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC Non-Uniform Windowed Decoding 2 Motivation and Contributions Ensemble Definition Decoder Design Simulations 3 Ongoing Work 4 Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 2 / 34
Outline Background and Previous Work 1 Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC Non-Uniform Windowed Decoding 2 Motivation and Contributions Ensemble Definition Decoder Design Simulations 3 Ongoing Work 4 Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 3 / 34
LDPC Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 4 / 34
Spatially-Coupled LDPC Codes Spatially-coupled (SC) LDPC codes are constructed by concatenating multiple LDPC codes into a single chain. SC-LDPC codes have capacity approaching performance and allow for low latency windowed decoding. [Iyengar ’12] Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 5 / 34
Outline Background and Previous Work 1 Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC Non-Uniform Windowed Decoding 2 Motivation and Contributions Ensemble Definition Decoder Design Simulations 3 Ongoing Work 4 Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 6 / 34
Multi-Dimensional SC-LDPC Codes Multi-dimensional (MD) SC codes are a subclass where copies of SC codes are coupled together, to create a code with outstanding finite and asymptotic performance. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 7 / 34
Benefits of MD-SC LDPC Codes Robustness to burst erasures [Ohashi ’13] Improved decoding thresholds [Liu ’15, Truhachev ’19] Improved reliability over parallel channels [Schmalen ’14] Continuous transmission of multiple codewords with good finite-length performance [Olmos ’17] Flexible finite-length design for removing detrimental objects to improve iterative decoding [Esfahanizadeh ’20, Hareedy ’20] New Result: Non-uniform windowed decoding for flexible decoder complexity Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 8 / 34
Previous Work: Low-Latency Multi-Dimensional Windowed Decoding Best known previous work on windowed decoding of MD-SC codes: All constituent SC codes have a window that move in unison along the MD-SC chain. [Esfahanizadeh ’20] Traditional uniform decoders allow a large latency reduction with almost the same performance; however, not taking full advantage of the MD-SC code structure maximize decoding flexibility. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 9 / 34
Outline Background and Previous Work 1 Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC Non-Uniform Windowed Decoding 2 Motivation and Contributions Ensemble Definition Decoder Design Simulations 3 Ongoing Work 4 Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 10 / 34
Outline Background and Previous Work 1 Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC Non-Uniform Windowed Decoding 2 Motivation and Contributions Ensemble Definition Decoder Design Simulations 3 Ongoing Work 4 Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 11 / 34
Non-Uniform Windowed Decoder Motivation In MD-SC codes, not all constituent codes are directly connected. Intuitively, codes that are not directly connected, may provide less information to decode the target VNs. Example: let (1 , 0) be the target VN section and all sections in the first column are already decoded. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 12 / 34
Non-Uniform Windowed Decoder Motivation In MD-SC codes, not all constituent codes are directly connected. Intuitively, codes that are not directly connected, may provide less information to decode the target VNs. Example: let (1 , 0) be the target VN section and all sections in the first column are already decoded. If the graph is strongly connected, the uniform windowed decoder is generally a good solution. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 12 / 34
Non-Uniform Windowed Decoder Motivation In MD-SC codes, not all constituent codes are directly connected. Intuitively, codes that are not directly connected, may provide less information to decode the target VNs. Example: let (1 , 0) be the target VN section and all sections in the first column are already decoded. If the sections are sparsely connected, select sections into the decoding window that are closer to the target VNs. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 12 / 34
Non-Uniform Windowed Decoder Motivation In MD-SC codes, not all constituent codes are directly connected. Intuitively, codes that are not directly connected, may provide less information to decode the target VNs. Example: let (1 , 0) be the target VN section and all sections in the first column are already decoded. Distance may not be a good metric to select sections in the window. For the target VNs, connections from CNs to VNs can be more important. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 12 / 34
Non-Uniform Windowed Decoder Motivation In MD-SC codes, not all constituent codes are directly connected. Intuitively, codes that are not directly connected, may provide less information to decode the target VNs. Example: let (1 , 0) be the target VN section and all sections in the first column are already decoded. Uniform windowed decoders do not fully exploit all these features. Thus, motivating the need for a non-uniform decoder. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 12 / 34
Key Contributions A new, more general MD-SC ensemble to allow for varying the coupling along different dimensions. A novel non-uniform shaped windowed decoder and its key design parameters. Optimization of design parameters using density evolution for the BEC channel and insight into decoder design. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 13 / 34
Outline Background and Previous Work 1 Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC Non-Uniform Windowed Decoding 2 Motivation and Contributions Ensemble Definition Decoder Design Simulations 3 Ongoing Work 4 Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 14 / 34
Quick Notation ( n ) B = n mod B . [ A ] � { 0 , 1 , . . . , A − 1 } [( L 1 , L 2 )] � { ( j , k ) : j ∈ [ L 1 ] , k ∈ [ L 2 ] } . Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 15 / 34
Multi-Dimensional SC-LDPC ensemble: Parameters Code Parameters d l , d r : degrees of VNs and CNs. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34
Multi-Dimensional SC-LDPC ensemble: Parameters Code Parameters d l , d r : degrees of VNs and CNs. L 1 : 1D coupling length, length of individual codes. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34
Multi-Dimensional SC-LDPC ensemble: Parameters Code Parameters d l , d r : degrees of VNs and CNs. L 1 : 1D coupling length, length of individual codes. L 2 : MD coupling length, number of codes coupled. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34
Multi-Dimensional SC-LDPC ensemble: Parameters Code Parameters d l , d r : degrees of VNs and CNs. L 1 : 1D coupling length, length of individual codes. L 2 : MD coupling length, number of codes coupled. γ 1 < L 1 : 1D coupling depth, maximum coupling distance along a constituent code. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34
Multi-Dimensional SC-LDPC ensemble: Parameters Code Parameters d l , d r : degrees of VNs and CNs. L 1 : 1D coupling length, length of individual codes. L 2 : MD coupling length, number of codes coupled. γ 1 < L 1 : 1D coupling depth, maximum coupling distance along a constituent code. γ 2 < L 2 : MD coupling depth, maximum coupling distance among constituent codes. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34
Multi-Dimensional SC-LDPC ensemble: Parameters Code Parameters d l , d r : degrees of VNs and CNs. L 1 : 1D coupling length, length of individual codes. L 2 : MD coupling length, number of codes coupled. γ 1 < L 1 : 1D coupling depth, maximum coupling distance along a constituent code. γ 2 < L 2 : MD coupling depth, maximum coupling distance among constituent codes. 0 ≤ T ≤ 1: edge density among constituent codes. Higher T means more edges among codes and vice versa. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34
Multi-Dimensional SC-LDPC ensemble: Parameters Code Parameters d l , d r : degrees of VNs and CNs. L 1 : 1D coupling length, length of individual codes. L 2 : MD coupling length, number of codes coupled. γ 1 < L 1 : 1D coupling depth, maximum coupling distance along a constituent code. γ 2 < L 2 : MD coupling depth, maximum coupling distance among constituent codes. 0 ≤ T ≤ 1: edge density among constituent codes. Higher T means more edges among codes and vice versa. M : number of VNs in a section. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34
Multi-Dimensional SC-LDPC Ensemble: Construction Assume we have sections that have M VNs and M ( d l / d r ) CNs each. 1 Figure: Example grid of L 1 = 4 and L 2 = 3. Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 17 / 34
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