Periodic Reporting for period 1 - 6G Channel Coding (Flexible Channel Coding for 6G Short Packet Communication)
Reporting period: 2022-10-01 to 2025-06-30
Summary of the context and overall objectives of the project
The upcoming sixth-generation (6G) mobile networks are poised to tackle global challenges like urbanization and the digital divide by aligning with the UN Sustainable Development Goals (SDGs), aiming to empower individuals, sense the environment, and strengthen global ecosystems by 2030. Achieving this demands capabilities far beyond current 5G standards, particularly for critical use cases like wireless factory automation and autonomous driving, which require ultra-high reliability, ultra-low latency, high-resolution localisation, and precise inter-device synchronicity. A key hurdle is the need for highly flexible and efficient short packet communications, as current channel coding and decoding techniques are insufficient for the short to moderate blocklengths vital for 6G. This project focused on developing novel rateless coding schemes and decoding algorithms specifically optimised for finite blocklengths, offering flexibility in code rates and blocklengths, and featuring low-complexity decoding to mitigate latency. The main objectives of this project are as follows:
• Objective 1: Develop flexible and granular rateless codes for finite blocklength channels.
• Objective 2: Reduce decoding complexity of the designed rateless codes.
• Objective 3: Minimise decoding iterations through novel approaches.
• Objective 1: Develop flexible and granular rateless codes for finite blocklength channels.
• Objective 2: Reduce decoding complexity of the designed rateless codes.
• Objective 3: Minimise decoding iterations through novel approaches.
Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far
1) PR Code Optimization:
Preliminary investigations into Primitive Rateless (PR) codes revealed the existence of code constructions exhibiting minimum Hamming weights that closely approximate theoretical limits, including the Gilbert-Varshamov bound. However, the systematic identification of such optimal PR codes posed a significant challenge. This study addressed this gap by proposing a novel construction methodology leveraging specific primitive polynomials. Our approach involves utilizing primitive polynomials that satisfy the criteria of Golomb Rulers. This specific selection guarantees the construction of a parity-check matrix for PR codes that is entirely devoid of 4-cycles. The absence of 4-cycles is a critical structural property that simplifies the identification of suitable primitive polynomials, and consequently, the efficient generation of corresponding PR codes. Our findings demonstrate that PR codes constructed via this method exhibit performance remarkably close to the finite-length bounds, particularly at low code rates.
This novel design offers several significant advantages:
• Reduced Search Space for Optimal Primitive Polynomials: The challenge of identifying the optimal primitive polynomial among Φ(2 k −1)/k possibilities of length k is significantly mitigated. By restricting the search to primitive polynomials that are also Golomb Rulers of length k, the search space is substantially reduced. This dramatically facilitates the discovery of high-performing PR codes.
• Enabling Low-Complexity Iterative Decoding: The construction method ensures that the generated PR codes possess at least one parity-check matrix without any 4-cycles. This structural property is highly advantageous for the application of low-complexity iterative decoders, such as belief propagation (BP) algorithms, thereby substantially boosting decoding efficiency. Further research into this aspect led to the development of various low-complexity decoders for PR codes.
• Broadened Applicability Across Code Rates: The proposed design enables the identification of effective PR codes not only for low-rate scenarios, but also for high-rate scenarios, an area previously underexplored.
PR codes inherently benefit from their simple encoder implementations, typically realized using linear feedback shift registers. The proposed scheme, detailed in Technical WP1, allows for the systematic design of PR codes for arbitrary rates and blocklengths. This methodology partially addresses the current research imperative for designing efficient rate-compatible codes, which are crucial for the development of future wireless communication systems, including 6G networks. This advancement contributes directly to addressing the flexibility and bit-granularity requirements for finite blocklength channel codes in these emerging systems.
2) Designing Threshold-based OSD Algorithm for PR Codes
Many channel codes, particularly those designed and optimized for short block lengths, requires complex decoders to achieve near-optimal performance. While there are several codes, like Polar and LDPC codes, which are currently being used in 5G standard, can be decoded with low-complexity decoders, such as successive cancellation list decoding and belief propagation decoder, respectively, their decoding performance are usually sub-optimal or suffer from large decoding latency. In current mobile standards, retransmission techniques, such are hybrid automatic request (HARQ) are being widely used to achieve high reliability. This is going to be used in future 6G too, and in particular for low-latency high reliability applications, where one to two retransmission are allowed to achieve the desired level of reliability. The major challenge for short packet communication involving HARQ is the received complexity, which currently requires the receiver to perform full decoding to identify non-decodable packets, and request for retransmission. This indeed add to the complexity. We proposed an early termination strategy to identify the decidability of the packets without performing full decoding, thus significantly reducing the decoding complexity. We trained a graph nueral network which provided a relatively high accuracy of the detection of packets decodability, which reduced the overall decoding complexity.
3) Designing Windowed-OSD for PR Codes
Beyond 5G (B5G) and 6G networks demand revolutionary advancements in wireless systems, particularly for ultra-reliable low-latency communications (URLLC) and massive machine-type communications (mMTC). URLLC needs sub-millisecond latency with incredibly low frame error rates (10^−5 to 10^−7 ), while mMTC focuses on reliable, energy-efficient transmission of short packets from countless low-power devices. Both require short block codes that deliver high reliability, low decoding latency, and manageable complexity. Recently, algebraic codes like Reed–Muller (RM) and BCH have regained attention due to their strong minimum distance properties. While universal decoding algorithms such as ordered statistics decoding (OSD) and guessing random additive noise decoding (GRAND) can approach maximum-likelihood (ML) performance, they become computationally intensive at the low SNRs and rates common in URLLC and mMTC. Belief propagation (BP) offers a lower-complexity, parallelizable alternative, but its effectiveness is tied to the sparsity and cycle structure of the parity-check matrix (PCM). Many algebraic codes have dense PCMs with performance-degrading short cycles, especially 4-cycles. Researchers have explored several improvements: adaptive BP (ABP), which permutes the PCM using Gaussian elimination per iteration; and structural design techniques like low-weight parity-check (LWPC) matrices or 4-cycle elimination using auxiliary variables. However, auxiliary-node designs often degrade performance, particularly in AWGN channels. More recently, the progressive row growth (PRG) scheme enabled 4-cycle-free PCM construction with better connectivity and minimal auxiliary overhead. Still, achieving near-ML performance often requires multiple-bases BP (MBBP) decoding, which is computationally expensive and hard to scale as it relies on constructing diverse PCMs from low-weight dual codewords. Primitive rateless (PR) codes offer a promising solution. Generated from m-sequences based on primitive polynomials over GF(2), PR codes naturally support rate compatibility and high flexibility. When constructed with Golomb ruler-supported primitive polynomials, PR codes can inherently produce 4-cycle-free PCMs without auxiliary variables, enabling efficient BP decoding.
In this work, we introduced a low-complexity double-bases BP decoder specifically designed for PR codes. This novel scheme achieves MBBP-level performance without the need for computationally expensive dual-codeword PCM construction. Our approach utilizes two PCMs: a sparse, 4-cycle-free PCM derived from a Golomb ruler, and a second matrix generated once by sorting channel log-likelihood ratios (LLRs) followed by Gaussian elimination. Unlike adaptive BP (ABP), our second PCM is fixed and computed only once, significantly reducing the overall complexity. Simulation results demonstrate that this method not only outperforms traditional MBBP but also offers a practical and scalable decoding solution for short PR codes.
4) Optimizing Stopping and Discarding Rules for the OSD Algorithm
For ultra-reliable low-latency communications (URLLC), achieving the strict latency and reliability goals requires short block-length codes with powerful error correction. Several codes, like extended Bose, Chaudhuri, and Hocquenghem (eBCH) codes, cyclic-redundancy-check-aided polar (CRC-Polar) codes, and polarization-adjusted convolutional (PAC) codes, have shown they can perform close to the theoretical limits (normal approximation bound) across different block lengths and signal-to-noise ratios (SNRs). However, meeting URLLC's latency demands also means using low-complexity decoders alongside high-performing codes. Recently, Ordered-Statistics Decoding (OSD) and Guessing Random Additive Noise Decoding (GRAND) have emerged as promising universal decoders for URLLC. These decoders can decode any linear block code, allowing for the use of well-established linear codes with flexible block lengths and rates in URLLC applications. Primitive Rateless (PR) codes can be designed for any block length and rate and have been shown to meet the Gilbert-Varshamov bound for rates below 0.5. Although PR codes have a slightly lower minimum Hamming weight compared to BCH codes, their fewer low-weight codewords lead to comparable block error rate (BLER) performance. These features, combined with their extremely simple encoding, make PR codes strong candidates for URLLC. They can be decoded using universal decoders like OSD or its variations, which offer lower complexity in terms of the number of test error patterns (TEPs). In this part of the project, we introduced the Boolean function construction of PR codes. We showed that decoding can be simplified to finding the dual basis of a given basis over {GF}(2^k), which involves matrix inversion over {GF}(2). Specifically, by using self-dual bases to construct high-rate PR codes (with message length k and codeword length n, where (n-k < k), the decoding process becomes even simpler. It requires inverting a t x t binary matrix with a complexity of {O}\left(t^3\right) binary operations, significantly less than the $\mathcal{O}\left(n(n-k)^2\right)$ operations needed for the conventional OSD algorithm's Gaussian elimination. Simulation results demonstrate that high-rate PR codes, when decoded with our proposed low-complexity method, achieve BLERs comparable to BCH codes across various SNRs and code rates. Unlike BCH codes, PR codes can be easily designed for any rate and block length by simply selecting a suitable primitive polynomial. Furthermore, our modified OSD decoder for PR codes requires significantly fewer test error patterns or codebook queries compared to soft guessing random additive noise decoder (SGRAND) and ordered-reliability bits GRAND (ORBGRAND).
Preliminary investigations into Primitive Rateless (PR) codes revealed the existence of code constructions exhibiting minimum Hamming weights that closely approximate theoretical limits, including the Gilbert-Varshamov bound. However, the systematic identification of such optimal PR codes posed a significant challenge. This study addressed this gap by proposing a novel construction methodology leveraging specific primitive polynomials. Our approach involves utilizing primitive polynomials that satisfy the criteria of Golomb Rulers. This specific selection guarantees the construction of a parity-check matrix for PR codes that is entirely devoid of 4-cycles. The absence of 4-cycles is a critical structural property that simplifies the identification of suitable primitive polynomials, and consequently, the efficient generation of corresponding PR codes. Our findings demonstrate that PR codes constructed via this method exhibit performance remarkably close to the finite-length bounds, particularly at low code rates.
This novel design offers several significant advantages:
• Reduced Search Space for Optimal Primitive Polynomials: The challenge of identifying the optimal primitive polynomial among Φ(2 k −1)/k possibilities of length k is significantly mitigated. By restricting the search to primitive polynomials that are also Golomb Rulers of length k, the search space is substantially reduced. This dramatically facilitates the discovery of high-performing PR codes.
• Enabling Low-Complexity Iterative Decoding: The construction method ensures that the generated PR codes possess at least one parity-check matrix without any 4-cycles. This structural property is highly advantageous for the application of low-complexity iterative decoders, such as belief propagation (BP) algorithms, thereby substantially boosting decoding efficiency. Further research into this aspect led to the development of various low-complexity decoders for PR codes.
• Broadened Applicability Across Code Rates: The proposed design enables the identification of effective PR codes not only for low-rate scenarios, but also for high-rate scenarios, an area previously underexplored.
PR codes inherently benefit from their simple encoder implementations, typically realized using linear feedback shift registers. The proposed scheme, detailed in Technical WP1, allows for the systematic design of PR codes for arbitrary rates and blocklengths. This methodology partially addresses the current research imperative for designing efficient rate-compatible codes, which are crucial for the development of future wireless communication systems, including 6G networks. This advancement contributes directly to addressing the flexibility and bit-granularity requirements for finite blocklength channel codes in these emerging systems.
2) Designing Threshold-based OSD Algorithm for PR Codes
Many channel codes, particularly those designed and optimized for short block lengths, requires complex decoders to achieve near-optimal performance. While there are several codes, like Polar and LDPC codes, which are currently being used in 5G standard, can be decoded with low-complexity decoders, such as successive cancellation list decoding and belief propagation decoder, respectively, their decoding performance are usually sub-optimal or suffer from large decoding latency. In current mobile standards, retransmission techniques, such are hybrid automatic request (HARQ) are being widely used to achieve high reliability. This is going to be used in future 6G too, and in particular for low-latency high reliability applications, where one to two retransmission are allowed to achieve the desired level of reliability. The major challenge for short packet communication involving HARQ is the received complexity, which currently requires the receiver to perform full decoding to identify non-decodable packets, and request for retransmission. This indeed add to the complexity. We proposed an early termination strategy to identify the decidability of the packets without performing full decoding, thus significantly reducing the decoding complexity. We trained a graph nueral network which provided a relatively high accuracy of the detection of packets decodability, which reduced the overall decoding complexity.
3) Designing Windowed-OSD for PR Codes
Beyond 5G (B5G) and 6G networks demand revolutionary advancements in wireless systems, particularly for ultra-reliable low-latency communications (URLLC) and massive machine-type communications (mMTC). URLLC needs sub-millisecond latency with incredibly low frame error rates (10^−5 to 10^−7 ), while mMTC focuses on reliable, energy-efficient transmission of short packets from countless low-power devices. Both require short block codes that deliver high reliability, low decoding latency, and manageable complexity. Recently, algebraic codes like Reed–Muller (RM) and BCH have regained attention due to their strong minimum distance properties. While universal decoding algorithms such as ordered statistics decoding (OSD) and guessing random additive noise decoding (GRAND) can approach maximum-likelihood (ML) performance, they become computationally intensive at the low SNRs and rates common in URLLC and mMTC. Belief propagation (BP) offers a lower-complexity, parallelizable alternative, but its effectiveness is tied to the sparsity and cycle structure of the parity-check matrix (PCM). Many algebraic codes have dense PCMs with performance-degrading short cycles, especially 4-cycles. Researchers have explored several improvements: adaptive BP (ABP), which permutes the PCM using Gaussian elimination per iteration; and structural design techniques like low-weight parity-check (LWPC) matrices or 4-cycle elimination using auxiliary variables. However, auxiliary-node designs often degrade performance, particularly in AWGN channels. More recently, the progressive row growth (PRG) scheme enabled 4-cycle-free PCM construction with better connectivity and minimal auxiliary overhead. Still, achieving near-ML performance often requires multiple-bases BP (MBBP) decoding, which is computationally expensive and hard to scale as it relies on constructing diverse PCMs from low-weight dual codewords. Primitive rateless (PR) codes offer a promising solution. Generated from m-sequences based on primitive polynomials over GF(2), PR codes naturally support rate compatibility and high flexibility. When constructed with Golomb ruler-supported primitive polynomials, PR codes can inherently produce 4-cycle-free PCMs without auxiliary variables, enabling efficient BP decoding.
In this work, we introduced a low-complexity double-bases BP decoder specifically designed for PR codes. This novel scheme achieves MBBP-level performance without the need for computationally expensive dual-codeword PCM construction. Our approach utilizes two PCMs: a sparse, 4-cycle-free PCM derived from a Golomb ruler, and a second matrix generated once by sorting channel log-likelihood ratios (LLRs) followed by Gaussian elimination. Unlike adaptive BP (ABP), our second PCM is fixed and computed only once, significantly reducing the overall complexity. Simulation results demonstrate that this method not only outperforms traditional MBBP but also offers a practical and scalable decoding solution for short PR codes.
4) Optimizing Stopping and Discarding Rules for the OSD Algorithm
For ultra-reliable low-latency communications (URLLC), achieving the strict latency and reliability goals requires short block-length codes with powerful error correction. Several codes, like extended Bose, Chaudhuri, and Hocquenghem (eBCH) codes, cyclic-redundancy-check-aided polar (CRC-Polar) codes, and polarization-adjusted convolutional (PAC) codes, have shown they can perform close to the theoretical limits (normal approximation bound) across different block lengths and signal-to-noise ratios (SNRs). However, meeting URLLC's latency demands also means using low-complexity decoders alongside high-performing codes. Recently, Ordered-Statistics Decoding (OSD) and Guessing Random Additive Noise Decoding (GRAND) have emerged as promising universal decoders for URLLC. These decoders can decode any linear block code, allowing for the use of well-established linear codes with flexible block lengths and rates in URLLC applications. Primitive Rateless (PR) codes can be designed for any block length and rate and have been shown to meet the Gilbert-Varshamov bound for rates below 0.5. Although PR codes have a slightly lower minimum Hamming weight compared to BCH codes, their fewer low-weight codewords lead to comparable block error rate (BLER) performance. These features, combined with their extremely simple encoding, make PR codes strong candidates for URLLC. They can be decoded using universal decoders like OSD or its variations, which offer lower complexity in terms of the number of test error patterns (TEPs). In this part of the project, we introduced the Boolean function construction of PR codes. We showed that decoding can be simplified to finding the dual basis of a given basis over {GF}(2^k), which involves matrix inversion over {GF}(2). Specifically, by using self-dual bases to construct high-rate PR codes (with message length k and codeword length n, where (n-k < k), the decoding process becomes even simpler. It requires inverting a t x t binary matrix with a complexity of {O}\left(t^3\right) binary operations, significantly less than the $\mathcal{O}\left(n(n-k)^2\right)$ operations needed for the conventional OSD algorithm's Gaussian elimination. Simulation results demonstrate that high-rate PR codes, when decoded with our proposed low-complexity method, achieve BLERs comparable to BCH codes across various SNRs and code rates. Unlike BCH codes, PR codes can be easily designed for any rate and block length by simply selecting a suitable primitive polynomial. Furthermore, our modified OSD decoder for PR codes requires significantly fewer test error patterns or codebook queries compared to soft guessing random additive noise decoder (SGRAND) and ordered-reliability bits GRAND (ORBGRAND).
Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)
6G technology is on the horizon, promising incredibly fast communication, especially in high-frequency bands, and will use ultra-massive MIMO technologies. To achieve this, we need significant changes to our network infrastructure and radio interfaces. A key area of innovation is the physical layer, where we need more flexibility in how we handle data blocks and coding rates. Current channel coding schemes in 5G won't be enough. We need new techniques that can deliver a lower error floor and better waterfall performance to handle fast-changing, low-capacity scenarios with short-packet communications. The new decoding algorithms must also work well with little to no channel state information (CSI) at the receiver. Our research proposed a solution to these challenges by designing universal channel coding and decoding techniques for 6G. Current systems are limited by using fixed-rate and fixed-length codes, which means devices need multiple encoder/decoder pairs to work in different conditions. This increases complexity and cost. Our proposed codes and decoders will solve this by allowing a single encoder/decoder to perform near-optimally across various channel conditions. This project will significantly advance the theory and design of rateless codes, which are crucial for meeting 6G requirements. We developed practical codes that can approach the GV bound for any blocklength. We also created cutting-edge decoding techniques, including fast implementations of ordered statistics decoders that use parallelization and efficient stopping rules.
Overview of the results:
1) Primitive rateless codes can be designed at any blocklength and rates, and by using primitive polynomials that their support satisfys the criteria for Golomb ruler, several parity check matrices can be easily designed to reduce the complexity of the decoder.
2) We proposed an early termination strategy to identify the decodability of the packets without performing full decoding, thus significantly reducing the decoding complexity. We trained a graph nueral network which provided a relatively high accuracy of the detection of packets decodability, which reduced the overall decoding complexity.
3) We introduced a low-complexity double-bases BP decoder specifically designed for PR codes. Our approach utilizes two PCMs: a sparse, 4-cycle-free PCM derived from a Golomb ruler, and a second matrix generated once by sorting channel log-likelihood ratios (LLRs) followed by Gaussian elimination. Unlike adaptive BP (ABP), our second PCM is fixed and computed only once, significantly reducing the overall complexity. Simulation results demonstrate that this method not only outperforms traditional MBBP but also offers a practical and scalable decoding solution for short PR codes.
4) We introduced the Boolean function construction of PR codes. We showed that decoding can be simplified to finding the dual basis of a given basis over {GF}(2^k). Simulation results demonstrate that high-rate PR codes, when decoded with our proposed low-complexity method, achieve BLERs comparable to BCH codes across various SNRs and code rates. Unlike BCH codes, PR codes can be easily designed for any rate and block length by simply selecting a suitable primitive polynomial.
The proposed approaches were investigated using numerical solutions. Further investigation is needed to better understand their efficiency for hardware implementation and real-world testing.
Overview of the results:
1) Primitive rateless codes can be designed at any blocklength and rates, and by using primitive polynomials that their support satisfys the criteria for Golomb ruler, several parity check matrices can be easily designed to reduce the complexity of the decoder.
2) We proposed an early termination strategy to identify the decodability of the packets without performing full decoding, thus significantly reducing the decoding complexity. We trained a graph nueral network which provided a relatively high accuracy of the detection of packets decodability, which reduced the overall decoding complexity.
3) We introduced a low-complexity double-bases BP decoder specifically designed for PR codes. Our approach utilizes two PCMs: a sparse, 4-cycle-free PCM derived from a Golomb ruler, and a second matrix generated once by sorting channel log-likelihood ratios (LLRs) followed by Gaussian elimination. Unlike adaptive BP (ABP), our second PCM is fixed and computed only once, significantly reducing the overall complexity. Simulation results demonstrate that this method not only outperforms traditional MBBP but also offers a practical and scalable decoding solution for short PR codes.
4) We introduced the Boolean function construction of PR codes. We showed that decoding can be simplified to finding the dual basis of a given basis over {GF}(2^k). Simulation results demonstrate that high-rate PR codes, when decoded with our proposed low-complexity method, achieve BLERs comparable to BCH codes across various SNRs and code rates. Unlike BCH codes, PR codes can be easily designed for any rate and block length by simply selecting a suitable primitive polynomial.
The proposed approaches were investigated using numerical solutions. Further investigation is needed to better understand their efficiency for hardware implementation and real-world testing.