Binary turbo product codes (TPCs) are powerful error-correcting codes constructed from short component codes. Traditionally, turbo product decoding passes log likelihood ratios (LLRs) between the component decoders, inherently losing information when bit correlation exists. Such correlation can arise exogenously from sources like intersymbol interference and endogenously during component code decoding. To preserve these correlations and improve performance, we propose turbo product decoding based on group probabilities. We theoretically predict mutual information and signal-to-noise ratio (SNR) gains of group over bit-probability decoding. To translate these theoretical insights to practice, we revisit non-binary TPCs that naturally support group-probability decoding. We show that any component list decoder that takes group probabilities as input and outputs block-wise soft-output can partially preserve bit correlation, which we demonstrate with symbol-level ORBGRAND combined with soft-output GRAND (SOGRAND). Our results demonstrate that group-probability-based turbo product decoding achieves SNR gains of up to 0.3 dB for endogenous correlation and 0.7 dB for exogenous correlation, compared to bit-probability decoding.

We introduce and analyze a discrete soft-decision channel called the linear reliability channel (LRC) in which the soft information is the rank-ordering of the received symbol reliabilities. We prove that the LRC is an appropriate approximation to a general class of binary-input, continuous-output channels when the noise variance is high. The central feature of the LRC is that its combinatorial nature allows for an extensive mathematical analysis of the channel and its corresponding hard- and soft-decision maximum-likelihood (ML) decoders. In particular, we establish explicit error exponents for ML decoding in the LRC when using random codes under both hard- and soft-decision decoding. This analysis allows for a direct, quantitative evaluation of the relative advantage of soft-decision decoding. The discrete geometry of the LRC is distinct from that of the BSC, which is characterized by the Hamming weight, offering a new perspective on code construction for soft-decision settings.

We present a novel method for error correction in the presence of fading channel estimation errors (CEE). When such errors are significant, considerable performance losses can be observed if the wireless transceiver is not adapted. Instead of refining the estimate by increasing the pilot sequence length or improving the estimation algorithm, we propose two new approaches based on Guessing Random Additive Noise Decoding (GRAND) decoders. The first method involves testing multiple candidates for the channel estimate located in the complex neighborhood around the original pilot-based estimate. All these candidates are employed in parallel to compute log-likelihood ratios (LLR). These LLRs are used as soft input to Ordered Reliability Bits GRAND (ORBGRAND). Posterior likelihood formulas associated with ORBGRAND are then computed to determine which channel candidate leads to the most probable codeword. The second method is a refined version of the first approach accounting for the presence of residual CEE in the LLR computation. The performance of these two techniques is evaluated for [128, 112] 5G NR CA-Polar and CRC codes. For the considered settings, block error rate (BLER) gains of several dBs are observed compared to cases where CEE is ignored.

Guessing Random Additive Noise Decoding (GRAND) and its variants, known for their near-maximum likelihood performance, have been introduced in recent years. One such variant, Segmented GRAND, reduces decoding complexity by generating only noise patterns that meet specific constraints imposed by the linear code. In this paper, we introduce a new method to efficiently derive multiple constraints from the parity check matrix. By applying a random invertible linear transformation and reorganizing the matrix into a tree structure, we extract up to$\log _{2} n$constraints, reducing the number of decoding queries while maintaining the structure of the original code for a code length of$n$. We validate the method through theoretical analysis and experimental simulations.

We propose a novel demodulation technique that leverages developments in guesswork-based forward error correction decoders and variable-length bit-to-symbol mappings. For most common channel models, the optimal modulation schemes are known to require nonuniform probability distributions over signal points, which presents practical challenges. An established way to map uniform binary sources to non-uniform symbol distributions is to assign a different number of bits to different constellation points. Doing so, however, means that erroneous demodulation at the receiver can lead to bit insertions or deletions, turning a channel with Hamming-type errors into an insertion-deletion channel. The demodulator we propose provides error detection and correction through the use of a low-overhead padding bit sequence. We evaluate the performance of the proposed demodulator in various channel models and various communication settings. We verify that the demodulator successfully corrects the insertion-deletion errors. Using the proposed demodulator, we study different constellation design schemes and how they behave in different channel conditions. Overall, we observe considerable gains that suggest, in some circumstances, one may improve the throughput while keeping the error rate the same.

A fully integrated hardware design of the universal maximum likelihood Guessing Random Additive Noise Decoding (GRAND) algorithm implemented in 40 nm CMOS is presented. It is shown how this integrated hard-detection decoder, which is designed to process component codes of up to 128 bits in length, can be extended to efficiently decode product codes as long as 16,384 bits using the Iterative GRAND (IGRAND) algorithm. Pipelined stages provide throughput gain and dynamic energy savings when channel noise conditions improve. The chip allows for decoding product codes with two distinct component codes due to its ability to interleave between two codebooks without any switch-over time. Measurements demonstrate the decoder’s accuracy and efficiency in decoding a broad selection of product codes, including the capacity-achieving random linear product codes. The chip consumes an average energy of 30.6 pJ/b with a latency of 1.04 μs when decoding the BCH(127,106,7) component code at 68 MHz from 1.1 V at a bit flip probability of 10-5. Using a single chip to decode a BCH(127,106,7)2product code which results in 16,129-bit code of rate 0.68, we demonstrate an average energy consumption of 61.2 pJ/b with an average latency of 265 μs for the same operating conditions.

We present a communication scheme using guessing random additive noise decoding (GRAND) to improve flexibility and reliability of the existing compressed error (CE) framework. The CE framework uses information feedback to construct follow-up transmissions by compressing previous noise realizations, offering high reliability at a code rate close to the forward channel capacity. The channel decoding algorithm GRAND allows us to efficiently maintain this performance in noisy feedback settings by shifting redundancy for forward message protection to the feedback channel. Our scheme, GRAND-CE, is therefore appropriate for cases where forward and feedback channel usage costs are asymmetric, e.g. uplink communications. GRAND-CE offers super-exponential error rate performance as channel use increases, with finite usage of a noisy feedback channel. Unlike the traditional forward error correction model, the receiver performs error correction encoding and the sender handles decoding. We also propose a technique for pipelining sequential transmissions to maintain fixed forward transmission length and good feedback channel coding performance.

We introduce an algorithm for approximating the codebook probability that is compatible with all successive cancellation (SC)-based decoding algorithms, including SC list (SCL) decoding. This approximation is based on an auxiliary distribution that mimics the dynamics of decoding algorithms with an SC decoding schedule. Based on this codebook probability and SCL decoding, we introduce soft-output SCL (SO-SCL) to generate both blockwise and bitwise soft-output (SO). Using that blockwise SO, we first establish that, in terms of both block error rate (BLER) and undetected error rate (UER), SO-SCL decoding of dynamic Reed-Muller (RM) codes significantly outperforms the CRC-concatenated polar codes from 5G New Radio under SCL decoding. Moreover, using SO-SCL, the decoding misdetection rate (MDR) can be constrained to not exceed any predefined value, making it suitable for practical systems. Proposed bitwise SO can be readily generated from blockwise SO via a weighted sum of beliefs that includes a term where SO is weighted by the codebook probability, resulting in a soft-input soft-output (SISO) decoder. Simulation results for SO-SCL iterative decoding of product codes and generalized LDPC (GLDPC) codes, along with information-theoretical analysis, demonstrate significant superiority over existing list-max and list-sum approximations.

We establish that a large, flexible class of long, high redundancy error correcting codes can be efficiently and accurately decoded with guessing random additive noise decoding (GRAND). Performance evaluation demonstrates that it is possible to construct simple product codes with lengths of approximately 200 to 4000 bits and rates between 0.2 and 0.8 that outperform low-density parity-check (LDPC) codes from the 5G New Radio standard in both AWGN and fading channels. The concatenated structure enables many desirable features, including: low-complexity hardware-friendly encoding and decoding; significant flexibility in length and rate through modularity; and high levels of parallelism in encoding and decoding that enable low latency. Central is the development of a method through which any soft-input (SI) GRAND algorithm can provide soft-output (SO) in the form of an accurate a-posteriori estimate of the likelihood that a decoding is correct or, in the case of list decoding, the likelihood that each element of the list is correct. The distinguishing feature of soft-output GRAND (SOGRAND) is the provision of an estimate that the correct decoding has not been found, even when providing a single decoding. Per-block SO can be converted into accurate per-bit SO by a weighted sum that includes a term for the SI. Implementing SOGRAND adds negligible computation and memory to the existing decoding process, and using it results in a practical, low-latency alternative to LDPC codes.

We present a framework that can exploit the tradeoff between the undetected error rate (UER) and block error rate (BLER) of polar-like codes. It is compatible with all successive cancellation (SC)-based decoding methods and relies on a novel approximation that we call codebook probability. This approximation is based on an auxiliary distribution that mimics the dynamics of decoding algorithms following an SC decoding schedule. Simulation results demonstrates that, in the case of SC list (SCL) decoding, the proposed framework outperforms the state-of-art approximations from Forney's generalized decoding rule for polar-like codes with dynamic frozen bits. In addition, dynamic Reed-Muller (RM) codes using the proposed generalized decoding significantly outperform CRC-concatenated polar codes decoded using SCL in both BLER and UER.

We propose guessing random additive noise decoding-aided macrosymbols (GRAND-AMs) as a nonorthogonal multiple access (NOMA) method that can detect, error correct, and decode multiple users with imperfect channel estimation, asynchronous transmission, and interference, which are all topics of concern for Internet of Things. GRAND-AM is a NOMA method that uses both joint multiuser detection and joint error correction decoding to handle multiple access interference (MAI). For the joint multiuser detector, we introduce the concept of a macrosymbol, which is constructed from the combination of all user symbols. For the error correction decoding component, we introduce multiple access channel (MAC) codes, which are codes that are used to split the channel rate between users and correct errors due to MAI. In this scheme, each user has their information bits encoded with independent MAC codes. We use a soft detection variant of GRAND, an efficient and practical decoding method that inverts noise effect sequences from a sequence of symbols to arrive at a codeword, to correct a sequence of macrosymbols, ensuring that all user codebooks are simultaneously satisfied. The joint detection and decoding of GRAND-AM can outperform time division multiple access (TDMA) by 10 dB with perfect channel estimation, and by 6 dB with imperfect channel estimation. Considering a more complete communication chain, when additional forward error correction is used along with the MAC code, the GRAND-AM method performs similarly to a same rate low-density parity-check-coded TDMA system.

To meet the Ultra Reliable Low Latency Communication (URLLC) needs of modern applications, there have been significant advances in the development of short error correction codes and corresponding soft detection decoders. A substantial hindrance to delivering low-latency is, however, the reliance on interleaving to break up omnipresent channel correlations to ensure that decoder input matches decoder assumptions. Consequently, even when using short codes, the need to wait to interleave data at the sender and de-interleave at the receiver results in significant latency that acts contrary to the goals of URLLC. Moreover, interleaving provably reduces capacity in channels with correlation, so that potential decoding performance is degraded. Here we introduce a variant of Ordered Reliability Bits Guessing Random Additive Noise Decoding (ORBGRAND), which we call ORBGRAND-Approximate Independence (ORBGRAND-AI), a soft-detection decoder that can decode any moderate redundancy code and overcomes the limitation of existing decoding paradigms by leveraging channel correlations and circumventing the need for interleaving. By leveraging correlation, not only is latency reduced, but error correction performance can be enhanced by multiple dB, while decoding complexity is also reduced, offering one potential solution for the provision of URLLC.

We establish that during the execution of any Guessing Random Additive Noise Decoding (GRAND) algorithm, an interpretable, useful measure of decoding confidence can be evaluated. This measure takes the form of a log-likelihood ratio (LLR) of the hypotheses that, should a decoding be found by a given query, the decoding is correct versus its being incorrect. That LLR can be used as soft output for a range of applications and we demonstrate its utility by showing that it can be used to confidently discard likely erroneous decodings in favor of returning more readily managed erasures. We show that feature can be used to compromise the physical layer security of short length wiretap codes by accurately and confidently revealing a proportion of a communication when code-rate is far above the Shannon capacity of the associated hard detection channel.

Guessing Random Additive Noise Decoding (GRAND) is a family of hard- and soft-detection error correction decoding algorithms that provide accurate decoding of any moderate redundancy code of any length. Here we establish a method through which any soft-input GRAND algorithm can provide soft output in the form of an accurate a posteriori estimate of the likelihood that a decoding is correct or, in the case of list decoding, the likelihood that the correct decoding is an element of the list. Implementing the method adds negligible additional computation and memory to the existing decoding process. The output permits tuning the balance between undetected errors and block errors for arbitrary moderate redundancy codes including CRCs.

Random jammers that overpower transmitted signals are a practical concern for many wireless communication protocols. As such, wireless receivers must be able to cope with standard channel noise and jamming (intentional or unintentional). To address this challenge, we propose a novel method to augment the resilience of the recent family of universal error-correcting GRAND algorithms. This method, called Erasure Decoding by Gaussian Elimination (EDGE), impacts the syndrome check block and is applicable to any variant of GRAND. We show that the proposed EDGE method naturally reverts to the original syndrome check function in the absence of erasures caused by jamming. We demonstrate this by implementing and evaluating GRAND-EDGE and ORBGRAND-EDGE. Simulation results, using a Random Linear Code (RLC) with a code rate of 105/128, show that the EDGE variants lower both the Block Error Rate (BLER) and the computational complexity by up to five order of magnitude compared to the original GRAND and ORBGRAND algorithms. We further compare ORBGRAND-EDGE to Ordered Statistics Decoding (OSD), and demonstrate an improvement of up to three orders of magnitude in the BLER.

Multiuser detection in multiple access channels is typically handled through individual detection of each user followed by individual error correction with a long error correcting code. In contrast, we introduce an alternative approach that uses guessing random additive noise decoding for macrosymbols, which is a joint multiuser detection method that works well with short error correcting codes. The macrosymbols are generated from the combination of the received symbols across all users, which are individually coded with short error correcting codes such as (8,4) cyclic redundancy check codes or (7,4) Hamming codes. Guessing random additive noise decoding with soft information on symbol level basis is then used to jointly correct both users on a macrosymbol level basis. The joint detection and error correction using the guessing random additive noise decoding algorithm aided macrosymbols method gives a 4 dB improvement in$E_{b}/N_{0}$over individual maximum likelihood multiuser detection and error correction.

For spectral efficiency, higher order modulation symbols confer information on more than one bit. As soft detection forward error correction decoders assume the availability of information at binary granularity, however, soft demappers are required to compute per-bit reliabilities from complex-valued signals. Here we show that the recently introduced universal soft detection decoder ORBGRAND can be adapted to work with symbol-level soft information, obviating the need for energy expensive soft demapping. We establish that doing so reduces complexity while retaining the error correction performance achieved with the optimal demapper.

For communications subject to correlated channel effects, noise recycling has recently been shown to enhance channel capacity with receiver-side-only changes. Using a taped-out chip, in a hard-detection scenario with guessing random additive noise decoding (GRAND), noise recycling has been established to both increase decoding accuracy and decrease decoding energy in single communication channels that employ interleavers. This paper presents results for the related soft-detection scenario by investigating noise recycling with an in-silicon realization of Ordered Reliability Bits Guessing Random Additive Noise Decoding (ORBGRAND). Measurements demonstrate that noise recycling leads to a significant reduction in the block error rate (BLER) and substantial improvements in latency and energy consumption by reducing the number of queries required for decoding. We also discuss dynamic lead channel selection for the soft detection scenario and show the importance of lead channel on overall decoding performance.

Guessing random additive noise decoding (GRAND) has recently demonstrated maximum-likelihood (ML) decoding performance on efficient, universal silicon realizations. Leveraging input bit-reliability soft information extracted from the channel and noise statistics, GRAND rank-orders and queries noise sequences in non-decreasing likelihood to recover code-words of arbitrary code-book structures. We consider soft-input, soft-output (SISO) GRAND that generates bit-reliability log-likelihood ratios (LLRs) via successive Euclidean-distance computations over a list of noise-recovered words. Noise guessing and list construction follow an ordered reliability bits GRAND (ORBGRAND) mechanism, the guess budget of which controls the performance and complexity trade-offs. The generated LLRs form enhanced a priori information that adapts noise-sequence ordering in a subsequent soft-GRAND iteration. We derive bounds on the achievable rates under per-realization and marginal input soft information and empirically study the achievable rates of SISO-GRAND. We also examine the complexity of the joint data detection and GRAND core, highlighting its superiority to conventional list-based detection schemes. SISO-ORBGRAND can outperform conventional sphere decoding in data detection and LLR generation; the corresponding channel-mismatched rates approximate ML decoding.

In a setting where the forward and feedback channel are noisy BSCs, we show how capacity is nearly achievable in a scheme with only source coding on the forward channel. In representative settings with noisy feedback, GRAND makes the scheme not only possible but practical. The sender transmits uncoded messages, and the receiver provides a noise effect guess as in GRAND, which is channel-encoded and sent to the receiver. With noiseless, finite-length feedback our scheme provides the type of super exponential error behavior associated in previous work with infinite-capacity feedback channels. With noisy feed-back, which is the more common setting in most systems, our scheme permits forward throughput that is effectively the same as in a noiseless feedback case. Moreover, the feedback channel usage remains limited. We propose a target error rate as a useful design parameter.

This work presents an interactive real-time demonstration of the first-integrated universal soft-detection decoder with an ultra-low energy consumption of 0.76pJ/bit and the lowest power of 4.9mW using Ordered Reliability Bits Guessing Random Additive Noise Decoding (ORBGRAND) [1]. The chip has a reconfigurable code length of 32 to 256 bits. The chip’s universality is demonstrated by decoding multimedia messages using different codebooks through an interactive Graphics User Interface (GUI). It is shown that the chip’s performance is independent of the codebook used and dynamically adapts to the channel noise conditions where lower energy is consumed as the Signal-to-Noise Ratio (SNR) of the channel improves.

Optimal modulation (OM) schemes for Gaussian channels with peak and average power constraints are known to require nonuniform probability distributions over signal points, which presents practical challenges. An established way to map uniform binary sources to non-uniform symbol distributions is to assign a different number of bits to different constellation points. Doing so, however, means that erroneous demodulation at the receiver can lead to bit insertions or deletions that result in significant binary error propagation. In this paper, we introduce a light-weight variant of Guessing Random Additive Noise Decoding (GRAND) to resolve insertion and deletion errors at the receiver by using a simple padding scheme. Performance evaluation demonstrates that our approach results in an overall gain in demodulated bit-error-rate of over 2 dB Eb/NO when compared to 128-Quadrature Amplitude Modulation (QAM). The GRAND-aided OM scheme outperforms coding with a low-density parity check code of the same average rate as that induced by our simple padding.

Guessing random additive noise decoding (GRAND) is a universal maximum-likelihood decoder that recovers codewords by guessing rank-ordered putative noise sequences and inverting their effect until one or more valid code-words are obtained. This work explores how GRAND can leverage additive-noise statistics and channel-state information in fading channels. Instead of computing per-bit reliability information in detectors and passing this information to the decoder, we propose leveraging the colored noise statistics following channel equalization as pseudo-soft information for sorting noise sequences. We investigate the efficacy of pseudo-soft information extracted from linear zero-forcing and minimum mean square error equalization when fed to a hardware-friendly soft-GRAND (ORBGRAND). We demonstrate that the proposed pseudo-soft GRAND schemes approximate the performance of state-of-the-art decoders of CA-Polar and BCH codes that avail of complete soft information. Compared to hard-GRAND, pseudo-soft ORBGRAND introduces up to 10 dB SNR gains for a target 10–3block-error rate.

Guessing random additive noise decoding (GRAND) is a maximum likelihood (ML) decoding method that identifies the noise effects corrupting code-words of arbitrary code-books. In a joint detection and decoding framework, this work demonstrates how GRAND can leverage crude soft information in received symbols and channel state information to generate, through guesswork, soft bit reliability outputs in log-likelihood ratios (LLRs). The LLRs are generated via successive computations of Euclidean-distance metrics corresponding to candidate noise-recovered words. Noting that the entropy of noise is much smaller than that of information bits, a small number of noise effect guesses generally suffices to hit a code-word, which allows generating LLRs for critical bits; LLR saturation is applied to the remaining bits. In an iterative (turbo) mode, the generated LLRs at a given soft-input, soft-output GRAND iteration serve as enhanced a priori information that adapts noise-sequence guess ordering in a subsequent iteration. Simulations demonstrate that a few turbo-GRAND iterations match the performance of ML-detection-based soft-GRAND in both AWGN and Rayleigh fading channels at a complexity cost that, on average, grows linearly (instead of exponentially) with the number of symbols.

Noise recycling is a recently proposed method that significantly enhances decoding performance when used for orthogonal channels impacted by correlated noise with only receiver side changes. In this paper, we establish that noise recycling can be applied in a single communication channel that is subject to temporally correlated noise by leveraging a standard matrix interleaver to create the effect of orthogonal channels. The proposed interleaved noise recycling technique works with any code, requires no sender-side alterations, and only minor changes to the receiver architecture. In a hard-detection scenario, we demonstrate noise recycling can enable an accurate estimate of continuous realization of noise without using any soft information, resulting in a gain of more than 2 dB in Block Error Rate (BLER). We use the first hardware implementation of Guessing Random Additive Noise Decoding (GRAND), a universal noise-centric decoder, to illustrate the advantages of noise recycling in hardware performance. At a correlation coefficient of 0.75, Eb/N0of 4 dB, a maximum of 36× decoding energy savings with a 12× reduction in latency is achieved using a BCH(127,113) code when GRAND is equipped with the proposed noise recycling.

We consider the problem of secure and reliable communication over a noisy multipath network. Previous work considering a noiseless version of our problem proposed a hybrid universal network coding cryptosystem (HUNCC). By combining an information-theoretically secure encoder together with partial encryption, HUNCC is able to obtain security guarantees, even in the presence of an all-observing eavesdropper. In this paper, we propose a version of HUNCC for noisy channels (N-HUNCC). This modification requires four main novelties. First, we present a network coding construction which is jointly, individually secure and error-correcting. Second, we introduce a new security definition which is a computational analogue of individual security, which we call individual indistinguishability under chosen ciphertext attack (individual IND-CCA1), and show that N-HUNCC satisfies it. Third, we present a noise based decoder for N-HUNCC, which permits the decoding of the encoded-then-encrypted data. Finally, we discuss how to select parameters for N-HUNCC and its error-correcting capabilities.

While many communications media, such as wireless and certain classes of wireline channels, typically lead to bursty errors, most decoders are designed assuming memoryless channels. Consequently, communication systems generally rely on interleaving over tens of thousands of bits to match decoder assumptions. Even for short high rate codes, awaiting sufficient data in interleaving and de-interleaving is a significant source of unwanted latency. We construct an extension to the recently proposed Guessing Random Additive Noise Decoding (GRAND) algorithm, which we call GRAND-MO for GRAND Markov Order. By foregoing interleaving and instead making use of the bursty nature of noise, low-latency communication is possible with block error rates outperforming their interleaved counterparts by a substantial margin. We establish that certain well-known binary codes with structured code-word patterns are ill-suited for use in bursty channels, but Random Linear Codes (RLCs) prove robust to correlated noise. We further demonstrate that by operating directly on modulated symbols rather than de-mapped bits, GRAND-MO achieves further performance and complexity gains by exploiting information that is lost in demodulation. As a result, GRAND-MO provides one potential solution for applications that require ultra-reliable low latency communication.

The design and implementation of error correcting codes has long been informed by two fundamental results: Shannon’s 1948 capacity theorem, which established that long codes use noisy channels most efficiently; and Berlekamp, McEliece, and Van Tilborg’s 1978 theorem on the NP-completeness of decoding linear codes. These results shifted focus away from creating code-independent decoders, but recent low-latency communication applications necessitate relatively short codes, providing motivation to reconsider the development of universal decoders. We introduce a scheme for employing binarized symbol soft information within Guessing Random Additive Noise Decoding, a universal hard detection decoder. We incorporate codebook-independent quantization of soft information to indicate demodulated symbols to be reliable or unreliable. We introduce two decoding algorithms: one identifies a conditional Maximum Likelihood (ML) decoding; the other either reports a conditional ML decoding or an error. For random codebooks, we present error exponents and asymptotic complexity, and show benefits over hard detection. As empirical illustrations, we compare performance with majority logic decoding of Reed-Muller codes, with Berlekamp-Massey decoding of Bose-Chaudhuri-Hocquenghem codes, with CA-SCL decoding of CA-Polar codes, and establish the performance of Random Linear Codes, which require a universal decoder and offer a broader palette of code sizes and rates than traditional codes.

We introduce Iterative GRAND (IGRAND), a universal product code decoder that applies iterative bounded distance decoding and decodes component codes using code-agnostic Guessing Random Additive Noise Decoding (GRAND). We empirically determine its accuracy and, based on GRAND hardware measurements, its complexity, showing gains over alternative algorithms. We prove that the class of product codes with random linear component codes, which IGRAND is capable of decoding, are capacity-achieving in hard-decision channels.

Modern applications are driving demand for ultra-reliable low-latency communications, rekindling interest in the performance of short, high-rate error correcting codes. To that end, here we introduce a soft-detection variant of Guessing Random Additive Noise Decoding (GRAND) called Ordered Reliability Bits GRAND that can decode any moderate redundancy block-code. For a code of n bits, it avails of no more than ⎾log2(n)⏋ bits of code-book-independent quantized soft detection information per received bit to determine an accurate decoding while retaining the original algorithm’s suitability for a highly parallelized implementation in hardware. ORBGRAND is shown to provide similar block error performance for codes of distinct classes (BCH, CA-Polar and RLC) with low complexity, while providing better block error rate performance than CA-SCL, a state of the art soft detection CA-Polar decoder.

CRC codes have long since been adopted in a vast range of applications. The established notion that they are suitable primarily for error detection can be set aside through use of the recently proposed Guessing Random Additive Noise Decoding (GRAND). Hard-detection (GRAND-SOS) and soft-detection (ORBGRAND) variants can decode any short, high-rate block code, making them suitable for error correction of CRC-coded data. When decoded with GRAND, short CRC codes have error correction capability that is at least as good as popular codes such as BCH codes, but with no restriction on either code length or rate.The state-of-the-art CA-Polar codes are concatenated CRC and Polar codes. For error correction, we find that the CRC is a better short code than either Polar or CA-Polar codes. Moreover, the standard CA-SCL decoder only uses the CRC for error detection and therefore suffers severe performance degradation in short, high rate settings when compared with the performance GRAND provides, which uses all of the CA-Polar bits for error correction.Using GRAND, existing systems can be upgraded from error detection to low-latency error correction without re-engineering the encoder, and additional applications of CRCs can be found in IoT, Ultra-Reliable Low Latency Communication (URLLC), and beyond. The universality of GRAND, its ready parallelized implementation in hardware, and the good performance of CRC as codes make their combination a viable solution for low-latency applications.

Two main problems arise in the Multiple Access Channel (MAC): interference from different users, and additive noise channel noise. Maximum A-Posteriori (MAP) joint decoding or successive interference cancellation are known to be capacity-achieving for the MAC when paired with appropriate codes. We extend the recently proposed Soft Guessing Random Additive Noise Decoder (SGRAND) to guess, using soft information, the effect of noise on the sum of users' transmitted codewords. Next, we manage interference by applying ZigZag decoding over the resulting putative noiseless MAC to obtain candidate codewords. Guessing continues until the candidate codewords thus obtained pertain to the corresponding users' codebooks. This MAC SGRAND decoder is a MAP decoder that requires no coordination between users, who can use arbitrary moderate redundancy short length codes of different types and rates.

To facilitate applications in IoT, 5G, and beyond, there is an engineering need to enable high-rate, low-latency communications. Errors in physical channels typically arrive in clumps, but most decoders are designed assuming that channels are memoryless. As a result, communication networks rely on interleaving over tens of thousands of bits so that channel conditions match decoder assumptions. Even for short high rate codes, awaiting sufficient data to interleave at the sender and de-interleave at the receiver is a significant source of unwanted latency. Using existing decoders with non-interleaved channels causes a degradation in block error rate performance owing to mismatch between the decoder's channel model and true channel behaviour.Through further development of the recently proposed Guessing Random Additive Noise Decoding (GRAND) algorithm, which we call GRAND-MO for GRAND Markov Order, here we establish that by abandoning interleaving and embracing bursty noise, low-latency, short-code, high-rate communication is possible with block error rates that outperform their interleaved counterparts by a substantial margin. Moreover, while most decoders are twinned to a specific code-book structure, GRANDMO can decode any code. Using this property, we establish that certain well-known structured codes are ill-suited for use in bursty channels, but Random Linear Codes (RLCs) are robust to correlated noise. This work suggests that the use of RLCs with GRAND-MO is a good candidate for applications requiring high throughput with low latency.

Maximum Likelihood (ML) decoding of forward error correction codes is known to be optimally accurate, but is not used in practice as it proves too challenging to efficiently implement. Here we propose a development of a previously described hard detection ML decoder called Guessing Random Additive Noise Decoding (GRAND). We introduce Soft GRAND (SGRAND), a ML decoder that fully avails of soft detection information and is suitable for use with any arbitrary high-rate, short-length block code. We assess SGRAND's performance on Cyclic Redundancy Check (CRC)-aided Polar (CA-Polar) codes, which will be used for all control channel communication in 5G New Radio (NR), comparing its accuracy with CRC-Aided Successive Cancellation List decoding (CA-SCL), a state-of-the-art soft-information decoder specific to CA-Polar codes.

We introduce Noise Recycling, a method that enhances decoding performance of channels subject to correlated noise without joint decoding. The method can be used with any combination of codes, code-rates and decoding techniques. In the approach, a continuous realization of noise is estimated from a lead channel by subtracting its decoded output from its received signal. This estimate is then used to improve the accuracy of decoding of an orthogonal channel that is experiencing correlated noise. In this design, channels aid each other only through the provision of noise estimates post-decoding. In a Gauss-Markov model of correlated noise, we constructively establish that noise recycling employing a simple successive order enables higher rates than not recycling noise. Simulations illustrate noise recycling can be employed with any code and decoder, and that noise recycling shows Block Error Rate (BLER) benefits when applying the same predetermined order as used to enhance the rate region. Finally, for short codes we establish that an additional BLER improvement is possible through noise recycling with racing, where the lead channel is not pre-determined, but is chosen on the fly based on which decoder completes first.

We recently introduced a noise-centric algorithm, Guessing Random Additive Noise Decoding (GRAND), that identifies a Maximum Likelihood (ML) decoding for arbitrary code-books. GRAND has the unusual property that its complexity decreases as code-book rate increases. Here we provide an extension to GRAND, soft-GRAND (SGRAND), that incorporates soft detection symbol reliability information and identifies a ML decoding in that context. In particular, we assume symbols received from the channel are declared to be error free or to have been potentially subject to additive noise. SGRAND inherits desirable properties of GRAND, including being capacity achieving when used with random code-books, and having a complexity that reduces as the code-rate increases.

Given a collection of strings, each with an associated probability of occurrence, the guesswork of each of them is their position in a list ordered from most likely to least likely, breaking ties arbitrarily. The guesswork is central to several applications in information theory: average guesswork provides a lower bound on the expected computational cost of a sequential decoder to decode successfully the transmitted message; the complementary cumulative distribution function of guesswork gives the error probability in list decoding; the logarithm of guesswork is the number of bits needed in optimal lossless one-to-one source coding; and the guesswork is the number of trials required of an adversary to breach a password protected system in a brute-force attack. In this paper, we consider memoryless string sources that generate strings consisting of independent and identically distributed characters drawn from a finite alphabet, and characterize their corresponding guesswork. Our main tool is the tilt operation on a memoryless string source. We show that the tilt operation on a memoryless string source parametrizes an exponential family of memoryless string sources, which we refer to as the tilted family of the string source. We provide an operational meaning to the tilted families by proving that two memoryless string sources result in the same guesswork on all strings of all lengths if and only if their respective categorical distributions belong to the same tilted family. Establishing some general properties of the tilt operation, we generalize the notions of weakly typical set and asymptotic equipartition property to tilted weakly typical sets of different orders. We use this new definition to characterize the large deviations for all atypical strings and characterize the volume of tilted weakly typical sets of different orders. We subsequently build on this characterization to prove large deviation bounds on guesswork and provide an accurate approximation of its probability mass function.

We introduce a new algorithm for realizing maximum likelihood (ML) decoding for arbitrary codebooks in discrete channels with or without memory, in which the receiver rank-orders noise sequences from most likely to least likely. Subtracting noise from the received signal in that order, the first instance that results in a member of the codebook is the ML decoding. We name this algorithm GRAND for Guessing Random Additive Noise Decoding. We establish that GRAND is capacity-achieving when used with random codebooks. For rates below capacity, we identify error exponents, and for rates beyond capacity, we identify success exponents. We determine the scheme's complexity in terms of the number of computations that the receiver performs. For rates beyond capacity, this reveals thresholds for the number of guesses by which, if a member of the codebook is identified, that it is likely to be the transmitted code word. We introduce an approximate ML decoding scheme where the receiver abandons the search after a fixed number of queries, an approach we dub GRANDAB, for GRAND with ABandonment. While not an ML decoder, we establish that the algorithm GRANDAB is also capacity-achieving for an appropriate choice of abandonment threshold, and characterize its complexity, error, and success exponents. Worked examples are presented for Markovian noise that indicate these decoding schemes substantially outperform the brute force decoding approach.

We study the central problem in data privacy: how to share data with an analyst while providing both privacy and utility guarantees to the user that owns the data. In this setting, we present an estimation-theoretic analysis of the privacy-utility trade-off (PUT). Here, an analyst is allowed to reconstruct (in a mean-squared error sense) certain functions of the data (utility), while other private functions should not be reconstructed with distortion below a certain threshold (privacy). We demonstrate how chi-square information captures the fundamental PUT in this case and provide bounds for the best PUT. We propose a convex program to compute privacy-assuring mappings when the functions to be disclosed and hidden are known a priori and the data distribution is known. We derive lower bounds on the minimum mean-squared error of estimating a target function from the disclosed data and evaluate the robustness of our approach when an empirical distribution is used to compute the privacy-assuring mappings instead of the true data distribution. We illustrate the proposed approach through two numerical experiments.

We introduce a new algorithm for Maximum Likelihood (ML) decoding for channels with memory. The algorithm is based on the principle that the receiver rank orders noise sequences from most likely to least likely. Subtracting noise from the received signal in that order, the first instance that results in an element of the code-book is the ML decoding. In contrast to traditional approaches, this novel scheme has the desirable property that it becomes more efficient as the code-book rate increases. We establish that the algorithm is capacity achieving for randomly selected code-books. When the code-book rate is less than capacity, we identify asymptotic error exponents as the block length becomes large. When the code-book rate is beyond capacity, we identify asymptotic success exponents. We determine properties of the complexity of the scheme in terms of the number of computations the receiver must perform per block symbol. Worked examples are presented for binary memoryless and Markovian noise. These demonstrate that block-lengths that offer a good complexity-rate tradeoff are typically smaller than the reciprocal of the bit error rate.

We explore properties and applications of the principal inertia components (PICs) between two discrete random variables X and Y. The PICs lie in the intersection of information and estimation theory, and provide a fine-grained decomposition of the dependence between X and Y. Moreover, the PICs describe which functions of X can or cannot be reliably inferred (in terms of MMSE), given an observation of Y. We demonstrate that the PICs play an important role in information theory, and they can be used to characterize information-theoretic limits of certain estimation problems. In privacy settings, we prove that the PICs are related to the fundamental limits of perfect privacy.

The problem of finding network codes for general connections is inherently difficult in capacity constrained networks. Resource minimization for general connections with network coding is further complicated. Existing methods for identifying solutions mainly rely on highly restricted classes of network codes, and are almost all centralized. In this paper, we introduce linear network mixing coefficients for code constructions of general connections that generalize random linear network coding for multicast connections. For such code constructions, we pose the problem of cost minimization for the subgraph involved in the coding solution and relate this minimization to a path-based constraint satisfaction problem (CSP) and an edge-based CSP. While CSPs are NP-complete in general, we present a path-based probabilistic distributed algorithm and an edge-based probabilistic distributed algorithm with almost sure convergence in finite time by applying communication free learning. Our approach allows fairly general coding across flows, guarantees no greater cost than routing, and shows a possible distributed implementation. Numerical results illustrate the performance improvement of our approach over existing methods.

The problem of finding network codes for general connections is inherently difficult. Resource minimization for general connections with network coding is further complicated. Existing methods for identifying solutions mainly rely on very restricted classes of network codes, and are almost all centralized. In this paper, we introduce linear network mixing coefficients for code constructions of general connections that generalize random linear network coding (RLNC) for multicast connections. For such code constructions, we pose the problem of cost minimization for the subgraph involved in the coding solution and relate this minimization to a Constraint Satisfaction Problem (CSP) which we show can be simplified to have a moderate number of constraints. While CSPs are NP-complete in general, we present a probabilistic distributed algorithm with almost sure convergence in finite time by applying Communication Free Learning (CFL). Our approach allows fairly general coding across flows, guarantees no greater cost than routing, and shows a possible distributed implementation. Numerical results illustrate the performance improvement of our approach over existing methods.

For general connections, the problem of finding network codes and optimizing resources for those codes is intrinsically difficult and little is known about its complexity. Most of the existing solutions rely on very restricted classes of network codes in terms of the number of flows allowed to be coded together, and are not entirely distributed. In this paper, we consider a new method for constructing linear network codes for general connections of continuous flows to minimize the total cost of edge use based on mixing. We first formulate the minimum-cost network coding design problem. To solve the optimization problem, we propose two equivalent alternative formulations with discrete mixing and continuous mixing, respectively, and develop distributed algorithms to solve them. Our approach allows fairly general coding across flows and guarantees no greater cost than any solution without inter-flow network coding.

Guesswork forms the mathematical framework for quantifying computational security subject to brute-force determination by query. In this paper, we consider guesswork subject to a per-symbol Shannon entropy budget. We introduce inscrutability rate as the asymptotic rate of increase in the exponential number of guesses required of an adversary to determine one or more secret strings. We prove that the inscrutability rate of any string-source supported on a finite alphabet χ, if it exists, lies between the per-symbol Shannon entropy constraint and log |χ|. We further prove that the inscrutability rate of any finite-order Markov string-source with hidden statistics remains the same as the unhidden case, i.e., the asymptotic value of hiding the statistics per each symbol is vanishing. On the other hand, we show that there exists a string-source that achieves the upper limit on the inscrutability rate, i.e., log |χ|, under the same Shannon entropy budget.

The guesswork problem was originally motivated by a desire to quantify computational security for single user systems. Leveraging recent results from its analysis, we extend the remit and utility of the framework to the quantification of the computational security of multi-user systems. In particular, assume that V users independently select strings stochastically from a finite, but potentially large, list. An inquisitor who does not know which strings have been selected wishes to identify U of them. The inquisitor knows the selection probabilities of each user and is equipped with a method that enables the testing of each (user, string) pair, one at a time, for whether that string had been selected by that user. Here, we establish that, unless U=V, there is no general strategy that minimizes the distribution of the number of guesses, but in the asymptote as the strings become long we prove the following: by construction, there is an asymptotically optimal class of strategies; the number of guesses required in an asymptotically optimal strategy satisfies a large deviation principle with a rate function, which is not necessarily convex, that can be determined from the rate functions of optimally guessing individual users' strings; if all users' selection statistics are identical, the exponential growth rate of the average guesswork as the string-length increases is determined by the specific Rényi entropy of the string-source with parameter (V-U+1)/(V-U+2), generalizing the known V=U=1 case; and that the Shannon entropy of the source is a lower bound on the average guesswork growth rate for all U and V, thus providing a bound on computational security for multi-user systems. Examples are presented to illustrate these results and their ramifications for systems design.

Consider the situation where a word is chosen probabilistically from a finite list. If an attacker knows the list and can inquire about each word in turn, then selecting the word via the uniform distribution maximizes the attacker's difficulty, its Guesswork, in identifying the chosen word. It is tempting to use this property in cryptanalysis of computationally secure ciphers by assuming coded words are drawn from a source's typical set and so, for all intents and purposes, uniformly distributed within it. By applying recent results on Guesswork, for i.i.d. sources it is this equipartition ansatz that we investigate here. In particular, we demonstrate that the expected Guesswork for a source conditioned to create words in the typical set grows, with word length, at a lower exponential rate than that of the uniform approximation, suggesting use of the approximation is ill-advised.

How hard is it to guess a password? Massey showed that a simple function of the Shannon entropy of the distribution from which the password is selected is a lower bound on the expected number of guesses, but one which is not tight in general. In a series of subsequent papers under ever less restrictive stochastic assumptions, an asymptotic relationship as password length grows between scaled moments of the guesswork and specific Rényi entropy was identified. Here, we show that, when appropriately scaled, as the password length grows, the logarithm of the guesswork satisfies a large deviation principle (LDP), providing direct estimates of the guesswork distribution when passwords are long. The rate function governing the LDP possesses a specific, restrictive form that encapsulates underlying structure in the nature of guesswork. Returning to Massey's original observation, a corollary to the LDP shows that expectation of the logarithm of the guesswork is the specific Shannon entropy of the password selection process.

We show that several important resource allocation problems in wireless networks fit within the common framework of constraint satisfaction problems (CSPs). Inspired by the requirements of these applications, where variables are located at distinct network devices that may not be able to communicate but may interfere, we define natural criteria that a CSP solver must possess in order to be practical. We term these algorithms decentralized CSP solvers. The best known CSP solvers were designed for centralized problems and do not meet these criteria. We introduce a stochastic decentralized CSP solver, proving that it will find a solution in almost surely finite time, should one exist, and also showing it has many practically desirable properties. We benchmark the algorithm's performance on a well-studied class of CSPs, random k-SAT, illustrating that the time the algorithm takes to find a satisfying assignment is competitive with stochastic centralized solvers on problems with order a thousand variables despite its decentralized nature. We demonstrate the solver's practical utility for the problems that motivated its introduction by using it to find a noninterfering channel allocation for a network formed from data from downtown Manhattan.

Rate control methodologies that are currently available in IEEE 802.11 network cards seriously underutilize network resources and, in addition, per-second throughputs suffer from high variability. In this paper, we introduce an algorithm, H-RCA, that overcomes these shortcomings, giving substantially higher, and less variable, throughput. The approach solely uses information already available at the driver-level to function and can be implemented on 802.11e commodity hardware. H-RCA's design objective is to minimize the average time each packet spends on the medium (including retries) in order to maximize total network throughput. It uses a development of a recently proposed estimation scheme to distinguish transmission failures due to collisions from those caused by channel noise. It employs an estimate of the packet loss ratio due to noise in assessing whether it is appropriate to change rate. We demonstrate experimentally that packet loss ratio is not necessarily a monotonic increasing function of rate; this is accounted for in H-RCA's design. As H-RCA statistically separates noise losses from those caused by collisions, ns-2 simulations show that it is robust to changing environments. H-RCA does not require specific hardware support nor any change to the IEEE 802.11 protocol. This point is substantiated with results from an experimental implementation.

The robustness to noise of the 802.11b/g 5.5 Mb/s and 11 Mb/s rates must be investigated experimentally as they cannot be predicted theoretically. In this paper we report on detailed outdoor and indoor measurements that lead us to the surprising conclusion that the 11 Mb/s 802.11g rate experiences fewer packet losses than the 6 Mb/s 802.11g rate at any given (symbol) SNR. This occurs due to the combination of modulation and physical layer coding schemes used by these rates and has serious implications for rate control algorithms. The practical implications of this, factoring in the interaction between packet loss and 802.11 MAC retries, is that 6 Mb/s is effectively redundant as a packet transmission rate if the 11 Mb/s rate is available.

We identify common hypotheses on which a large number of distinct mathematical models of WLANs employing IEEE 802.11 are founded. Using data from an experimental test bed and packet-level ns-2 simulations, we investigate the veracity of these hypotheses. We demonstrate that several of these assumptions are inaccurate and/or inappropriate. We consider hypotheses used in the modeling of saturated and unsaturated 802.11 infrastructure mode networks, saturated 802.11e networks, and saturated and unsaturated 802.11s mesh networks. In infrastructure mode networks, we find that even for small numbers of stations, common hypotheses hold true for saturated stations and also for unsaturated stations with small buffers. However, despite their widespread adoption, common assumptions used to incorporate station buffers are erroneous. This raises questions about the predictive power of all models based on these hypotheses. For saturated 802.11e models that treat differences in arbitration interframe space (AIFS), we find that the two fundamental hypotheses are reasonable. For 802.11s mesh networks, we find that assumptions are appropriate only if stations are lightly loaded and are highly inappropriate if they are saturated. In identifying these flawed suppositions, this work identifies areas where mathematical models need to be revisited and revised if they are to be used with confidence by protocol designers and WLAN network planners.

To extend established concepts of fair resource allocation in wired networks to wireless networks, wired model assumptions must be adapted to be relevant for wireless networks as for example, in wireless networks losses due to environmental conditions may occur even in the absence of queueing congestion. Thus fundamental questions of the existence and uniqueness of fair rate allocations must be reconsidered. We treat wireless networks characterized by lossy channels, spatial channel reuse, multiple routes and multiple frequencies. We establish the existence and uniqueness of utility fair and max-min fair solutions and that, as loss rates decrease, fair allocations converge to the loss-less ones.

As WLANs employing IEEE 802.11 have become pervasive, many analytic models for predicting their performance have been developed in recent years. Due to the complicated nature of the 802.11 MAC operation, approximations must be made to enable tractable mathematical models. In this article, through simulation we investigate the veracity of the approximations shared by many models that have been developed starting with the fundamental hypotheses in Bianchipsilas (1998 and 2000) seminal papers. We find that even for small numbers of station these assumptions that hold true for saturated stations (those that always have a packet to send) and for unsaturated stations with small buffers. However, despite their widespread adoption, we find that the commonly adopted assumptions that are used to incorporate station buffers are not appropriate. This raises questions about the predictive power of models based on these hypotheses.

This paper introduces a finite load multi-class 802.11e EDCF model that is simple enough to be explicitly solvable. The model is nevertheless flexible enough to model the impact of 802.11e parameters on the prioritization of realistic traffic. We emphasize that a modeling framework which allows nonsaturated sources is essential in the study of realistic traffic. We apply the model to a situation of practical interest: competing TCP flows in an infrastructure network. The model allows us to make a principled selection of 802.11e parameters to resolve problems highlighted in this scenario. Model predictions and parameter selections are validated against simulation and experiment. The model is shown to be accurate and the parameters effective.