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A Graphical Correlation-Based Method for Counting the Number of Global 8-Cycles on the SCRAM Three-Layer Tanner Graph

Sally Nafie, Joerg Robert, Albert Heuberger

TL;DR

A novel graphical approach that derives a lower bound on the shortest cycle length of an arbitrary SCRAM Tanner graph and presents a novel graphical method that counts the number of cycles of length that corresponds to the girth.

Abstract

This paper presents a novel graphical approach that counts the number of global 8-cycles on the SCRAM three-layer Tanner graph. SCRAM, which stands for Slotted Coded Random Access Multiplexing, is a joint decoder that is meets challenging requirements of 6G. At the transmitter side, the data of the accommodated users is encoded by Low Density Parity Check (LDPC) codes, and the codewords are transmitted over the shared channel by means of Slotted ALOHA. Unlike the state-of-the-art sequential decoders, the SCRAM decoder jointly resolves collisions and decodes the LDPC codewords, in a similar analogy to Belief Propagation on a three-layer Tanner graph. By leveraging the analogy between the two-layer Tanner graph of conventional LDPC codes and the three-layer Tanner graph of SCRAM, the well-developed analysis tools of classical LDPC codes could be utilized to enhance the performance of SCRAM. In essence, the contribution of this paper is three-fold; First it proposes the methodology to utilize these tools to assess the performance of SCRAM. Second, it derives a lower bound on the shortest cycle length of an arbitrary SCRAM Tanner graph. Finally, the paper presents a novel graphical method that counts the number of cycles of length that corresponds to the girth.

A Graphical Correlation-Based Method for Counting the Number of Global 8-Cycles on the SCRAM Three-Layer Tanner Graph

TL;DR

A novel graphical approach that derives a lower bound on the shortest cycle length of an arbitrary SCRAM Tanner graph and presents a novel graphical method that counts the number of cycles of length that corresponds to the girth.

Abstract

This paper presents a novel graphical approach that counts the number of global 8-cycles on the SCRAM three-layer Tanner graph. SCRAM, which stands for Slotted Coded Random Access Multiplexing, is a joint decoder that is meets challenging requirements of 6G. At the transmitter side, the data of the accommodated users is encoded by Low Density Parity Check (LDPC) codes, and the codewords are transmitted over the shared channel by means of Slotted ALOHA. Unlike the state-of-the-art sequential decoders, the SCRAM decoder jointly resolves collisions and decodes the LDPC codewords, in a similar analogy to Belief Propagation on a three-layer Tanner graph. By leveraging the analogy between the two-layer Tanner graph of conventional LDPC codes and the three-layer Tanner graph of SCRAM, the well-developed analysis tools of classical LDPC codes could be utilized to enhance the performance of SCRAM. In essence, the contribution of this paper is three-fold; First it proposes the methodology to utilize these tools to assess the performance of SCRAM. Second, it derives a lower bound on the shortest cycle length of an arbitrary SCRAM Tanner graph. Finally, the paper presents a novel graphical method that counts the number of cycles of length that corresponds to the girth.

Paper Structure

This paper contains 20 sections, 10 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. SCRAM Preliminaries
  3. System Model
  4. Three-Layer Tanner Graph Representation
  5. Iterative Joint SCRAM Decoding
  6. Check Nodes to Variable Nodes
  7. SA Check Nodes to Variable Nodes
  8. LDPC Check Nodes to Variable Nodes
  9. Variable Nodes to Check Nodes
  10. Cycle-Profile of SCRAM
  11. Cycle-Profile of Classical LDPC Codes
  12. Full-Cycle Algorithm
  13. Half-Cycle Algorithm
  14. Cycle Profile Analysis of a Classical LDPC Code
  15. SCRAM as a Hybrid Matrix
  16. ...and 5 more sections

Figures (10)

  • Figure 1: Three-Layer Tanner graph example of SCRAM system with $N_{u}=4$ users, each tranmitting $n_{n_{u}}=6$ LDPC encoded symbols over a system with $N_{s}=12$ SA slots
  • Figure 2: Example of Full-Cycle Algorithm on a Tanner graph of $n=6$ variable nodes and $m=5$ check nodes, counting the cycles that pass through variable node $v_{3}$
  • Figure 3: Hybrid Matrix of a SCRAM system with $N_{u}=4$ Users, each transmitting $n_{n_{u}}=6$ LDPC encoded symbols over a system with $N_{s}=12$ SA slots, by means of Random Access
  • Figure 4: Track of flow of beliefs at $t=0,1,2$, on a SCRAM graph with $N_{u}=4$ users, $n_{n_{u}}=6$ symbols, and $N_{s}=12$ slots
  • Figure 5: Track of flow of beliefs at $t=3,4,5$, Case 1, on a SCRAM graph with $N_{u}=4$ users, $n_{n_{u}}=6$ symbols, and $N_{s}=12$ slots
  • ...and 5 more figures