Patterned Reed–Muller Sequences with Outer A-Channel Codes and Projective Decoding for Slot-Controlled Unsourced Random Access
Abstract
:1. Introduction
1.1. Main Contributions of the Paper
- This paper designs a common codebook that optimizes URA energy efficiency performance by embedding zero bits in the second-order Reed–Muller sequences in accordance with a specific principle, which is the simplified version proposed by Pllaha et al. (2022) [37,38,39], and we denote it as a patterned Reed–Muller (PRM) code.
- Instead of the available common codebook that uses Reed–Muller (RM) properties in the binary domain, we explore its exclusive natures in the complex domain. In detail, we prove the algebraic and geometric properties of the second-order Reed–Muller sequence in the complex field, and related theories for PRM codes are also proven.
- A projective decoder is proposed for the PRM sequence and the fundamental theory for proposing such precise detection algorithm stems from its geometry property.
- The dependencies enlightened by the patterned property of PRM sequences prescribe how the information messages are mapped to the elements in a pool of slot-pattern controls (SPCs). The information message guides a single user to select the corresponding SPC from the pool and users randomly select SPCs as their transmission criteria to reduce collision chance.
- The factors affecting the reliability of the PRM detection are discussed. As a result of our simulations, we conclude that the proposed slot-pattern-control PRM-based scheme offers significant advantages in terms of error probability.
- Instead of the outer tree code proposed by Amalladinne et al. (2020) [19], we couple the proposed slot-pattern-control PRM-based scheme with two practical list recoverable codes. The first is a modification of the tree code called t-tree code and the second is based on the Reed–Solomon codes and Guruswami–Sudan list decoding algorithm. The optimal setups are determined to minimize SNR by optimizing the inner and outer codes jointly. In regimes of practical interest, the proposed scheme compares favorably with benchmark schemes regarding the energy-per-bit requirement to meet a target error probability as well as the number of accommodated active users in the system.
1.2. Notation
2. System Model
3. Reed–Muller Sequences
3.1. Binary RM Codes
3.2. Geometry of Complex RM Sequence
4. Expansion of Complex Reed–Muller Sequences
4.1. Patterned Reed–Muller Codes
- The evidence demonstrates that, given a sequence of length , PRM codes may increase the origin capacity of while maintaining the code distance and the cardinality of the PRM set is equal to
- In PRM sequence construction, the RM is not just inserted into a -length sequence, but has a sign attached (Equation (12) illustrates this). The sign comes from the part, i.e., the constraint of means the post- bits should be equal, which leads to the two identical vector multiplication (modulo 4) equal to the weight of or ;
- The vector and matrix serve as the determinants of the non-zero RM part, where only the upper matrix is valid. Moreover, the subspace and determine the “patterned” form.
4.2. Geometry Property of PRM sequence
4.3. The Proposed Geometry-Based PRM Detection
Algorithm 1: Estimation of single PRM sequence |
5. Unsourced Random Access Scheme Using Patterned Reed–Muller Sequence
5.1. Transmitter
5.1.1. Transmitter Design
5.1.2. The Construction of Slot-Pattern-Control Pool
5.2. Slot-Based PRM Detection and Reed–Solomon List Recovery Decoding
Algorithm 2: PRM reconstruction in a multi-user scenario |
6. Performance Analysis
The PRM Distribution for a Single Slot
7. Simulation Results
7.1. The Distribution of Slot-Based PRM Sequences
- PRM sequences are distributed more evenly when active users employ different SPCs. If collisions occur, more codewords will overlap at one slot, resulting in a rise in multi-user interference (MUI) and an increased probability of failure detection.
- Figure 6 illustrates that the “no collision” case is no longer valid when .
7.2. The Overall Performance of the SPC-Based CCS for URA System
7.2.1. t-Tree Code as the Outer Code
- PRM sequence for a given rank suffices for a user’s message delivery, indicating that the PRM codebook is highly spectral efficient due to its large sequence space.
- The outer-code length is used for and for . Substituting and into (38), we observe that the latter case () performs relatively poorly since it has a greater number of simultaneous appearances, which leads to inner-code failure at for and, for , . This result is consistent with Figure 6.
- For the curves of , the “PRM-tree” scheme corrects the case of the “t-tree code” scheme in which the overall performance degrades as t increases, i.e., an outer code with a larger t performs better on the condition that the inner code has the same length and the required path number is sufficient (“PRM-tree” schemes use paths and paths for “t-tree code”).
7.2.2. PRM-Based Reed–Solomon Scheme
- The minimum exceeds when the user count reaches 220.
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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t | G | J | |||||
---|---|---|---|---|---|---|---|
19 | 9 | 10 | |||||
15 | 7 | 8 | − | − | |||
15 | 7 | 8 | − | − |
Parameter Description | Specific Value |
---|---|
Transmit a message of size, B | 100 bits |
PRM sequence length, | |
The length of RS codes, | |
The number of slots, | |
The number of complex channel uses, T | |
The capacity of slot-occupation pool, | |
The length of slot-occupation control, | |
G-ary | |
J-ary | |
The capacity of PRM codebook, | |
The code rate |
Parameter Description | Specific Value |
---|---|
Transmit a message of size, B | 100 bits |
PRM sequence length, | |
The length of RS codes, | |
The number of slots, | |
The number of complex channel uses, T | |
The capacity of slot-occupation pool, | |
The length of slot-occupation control, | 9 |
G-ary | 15 |
J-ary | 6 |
The capacity of PRM codebook, | |
The code rate of RS |
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Xie, W.; Zhang, H. Patterned Reed–Muller Sequences with Outer A-Channel Codes and Projective Decoding for Slot-Controlled Unsourced Random Access. Sensors 2023, 23, 5239. https://doi.org/10.3390/s23115239
Xie W, Zhang H. Patterned Reed–Muller Sequences with Outer A-Channel Codes and Projective Decoding for Slot-Controlled Unsourced Random Access. Sensors. 2023; 23(11):5239. https://doi.org/10.3390/s23115239
Chicago/Turabian StyleXie, Wenjiao, and Huisheng Zhang. 2023. "Patterned Reed–Muller Sequences with Outer A-Channel Codes and Projective Decoding for Slot-Controlled Unsourced Random Access" Sensors 23, no. 11: 5239. https://doi.org/10.3390/s23115239
APA StyleXie, W., & Zhang, H. (2023). Patterned Reed–Muller Sequences with Outer A-Channel Codes and Projective Decoding for Slot-Controlled Unsourced Random Access. Sensors, 23(11), 5239. https://doi.org/10.3390/s23115239