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We consider the problem of communicating over a channel that randomly “tears” the message block into small pieces of different sizes and shuffles them. For the binary torn-paper channel with block length $n$ and pieces of length ${\mathrm{ Geometric}}(p_{n})$ , we characterize the capacity as $C = e^{-\alpha }$ , where $\alpha = \lim _{n\to \infty } p_{n} \log n$. Our results show that the case of ${\mathrm{ Geometric}}(p_{n})$ -length fragments and the case of deterministic length- $(1/p_{n})$ fragments are qualitatively different and, surprisingly, the capacity of the former is larger. Intuitively, this is due to the fact that, in the random fragments case, large fragments are sometimes observed, which boosts the capacity. [ABSTRACT FROM AUTHOR]

This report describes The University of Texas Student Cluster Competition team’s effort to reproduce the results of “MemXCT: memory-centric X-ray CT reconstruction with massive parallelization” (Hidayetoğlu et al., 2019). The article details a new memory-centric approach that reconstructs X-ray computed tomography (XCT) from noisy raw data. In our reproduction experiments, we utilized Microsoft Azure’s CycleCloud tool to provision, orchestrate, and manage our computing cluster in the cloud. In particular, we scheduled and benchmarked reconstruction workloads using Azure’s CPU-based HC44rs and GPU-based NC12s v2 virtual machine (VM) types to evaluate the scalability properties of the reconstruction approach and the performance differences between architectures. The HC44rs VMs contained 44 Intel Xeon Platinum cores, while the NC12s v2 VM was equipped with two NVIDIA P100 GPUs. We used a recent version of Intel’s compiler stack with the MKL library for our CPU code along with CUDA 11.1 on GPUs. Overall, our results confirm the findings of the original article, demonstrating similar acceleration on GPUs and scalability properties on CPUs. Digital artifacts from these experiments are available at: 10.5281/zenodo.5598108 [ABSTRACT FROM AUTHOR]

This report analyzes the reproducibility of the paper “MemXCT: Memory-Centric X-ray CT Reconstruction with Massive Parallelization” by Hidayetoğlu et al. in the cloud as part of the SC20 Virtual Student Cluster Competition (VSCC). To reproduce the results from the original work, the ETH Zürich SC20 VSCC team performed a series of CT reconstructions and performed a scaling study using three provided sinograms. All experimental runs were performed during the SC20 VSCC on an HPC cluster hosted in the Microsoft Azure CycleCloud. In this paper, we describe discrepancies in results as a factor of the differences in experiment environments and insufficient parameter tuning. We successfully reproduce the single device performance and partially reproduce the strong scaling behavior. Digital artifacts from these experiments are available at: 10.5281/zenodo.5598108. [ABSTRACT FROM AUTHOR]

The number-theoretic code is a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts’ codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo $r$. [ABSTRACT FROM AUTHOR]

Linear codes achieving or nearly achieving the Singleton bound are interesting in both theory and practice. The objective of this paper is to construct several infinite families of MDS, near MDS and almost MDS codes from some special cyclic subgroups of ${\mathbb {F}}_{q^{2}}^{*}$. To this end, the augmentation and extension techniques are used. The codes in this paper have flexible parameters and their lengths could be large. The minimum linear locality of the codes constructed in this paper is also studied. Some infinite families of optimal linearly locally recoverable codes are obtained. Besides, some codes in this paper are proved to be proper for error detection. [ABSTRACT FROM AUTHOR]

In general coding theory, we often assume that error is observed in transferring or storing encoded symbols, while the process of encoding itself is error-free. Motivated by recent applications of coding theory, in this paper, we consider the case where the process of encoding is distributed and prone to error. We introduce the problem of distributed encoding, comprised of a set of $K \in \mathbb {N}$ isolated source nodes and $N \in \mathbb {N}$ encoding nodes. Each source node has one symbol from a finite field, which is sent to each of the encoding nodes. Each encoding node stores an encoded symbol from the same field, as a function of the received symbols. However, some of the source nodes are controlled by the adversary and may send different symbols to different encoding nodes. Depending on the number of the adversarial nodes, denoted by $\beta \in \mathbb {N}$ , and the cardinality of the set of symbols that each one generates, denoted by $v \in \mathbb {N}$ , the process of decoding from the encoded symbols could be impossible. Assume that a decoder connects to an arbitrary subset of $t \in \mathbb {N}$ encoding nodes and wants to decode the symbols of the honest nodes correctly, without necessarily identifying the sets of honest and adversarial nodes. An important characteristic of a distributed encoding system is $t^{*} \in \mathbb {N}$ , the minimum of such $t$ , which is a function of $K$ , $N$ , $\beta $ , and $v$. In this paper, we study the distributed linear encoding system, i.e. one in which the encoding nodes use linear coding. We show that $t^{*}_{\textrm {Linear}}=K+2\beta (v-1)$ , if $N\ge K+2\beta (v-1)$ , and $t^{*}_{\textrm {Linear}}=N$ , if $N\le K+2\beta (v-1)$. In order to achieve $t^{*}_{\textrm {Linear}}$ , we use random linear coding and show that in any feasible solution that the decoder finds, the messages of the honest nodes are decoded correctly. In order to prove the converse of the fundamental limit, we show that when the adversary behaves in a particular way, it can always confuse the decoder between two feasible solutions that differ in the message of at least one honest node. [ABSTRACT FROM AUTHOR]

This paper provides a comprehensive performance comparison between a current robust military waveform; namely, MIL-STD-188-110D, Waveform 0, and a filter bank multicarrier spread-spectrum (FBMC-SS) waveform proposed for communications through ionospheric/skywave HF channels. Waveform 0 is effectively a direct sequence spread spectrum waveform that uses Walsh multi-codes to enhance the information transmission rate. It may thus be referred to as Walsh-DSSS. FBMC-SS, on the other hand, makes use of filter banks to provide excellent performance when the received signal is subject to partial band interference. Successful application of FBMC-SS for communications across skywave HF channels has been previously demonstrated, both theoretically and through experimental work. However, very little has been done to contrast FBMC-SS against Walsh-DSSS. The goal of this paper is to first add new features to FBMC-SS to bring it on par with Walsh-DSSS. These features include: (i), introduction of multi-codes that achieve a comparable (or better) data rate to Walsh-DSSS; and (ii), addition of a scrambling step applied to the multi-codes to make the receiver detection robust against widely spread multipaths. With this established, in the second part of the paper, we examine the performance of the developed FBMC-SS against Walsh-DSSS when both are applied for communications across skywave HF channels. The two waveforms are compared both through a theoretical study and through experimental works across several skywave channels ranging from hundreds to thousands of kilometers. [ABSTRACT FROM AUTHOR]

A functional $k$ -batch code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. Any multiset request of size $k$ of linear combinations (or requests) of the information bits can be recovered by $k$ disjoint subsets of the servers. The goal under this paradigm is to find the minimum number of servers for given values of $s$ and $k$. A recent conjecture states that for any $k=2^{s-1}$ requests the optimal solution requires $2^{s}-1$ servers. This conjecture is verified for $s \leqslant 5$ but previous work could only show that codes with $n=2^{s}-1$ servers can support a solution for $k=2^{s-2} + 2^{s-4} + \left \lfloor{ \frac { 2^{s/2}}{\sqrt {24}} }\right \rfloor $ requests. This paper reduces this gap and shows the existence of codes for $k=\lfloor \frac {5}{6}2^{s-1} \rfloor - s$ requests with the same number of servers. Another construction in the paper provides a code with $n=2^{s+1}-2$ servers and $k=2^{s}$ requests, which is an optimal result. These constructions are mainly based on extended Simplex codes and equivalently provide constructions for parallel Random I/O (RIO) codes. [ABSTRACT FROM AUTHOR]

Quantum readout provides a general framework for formulating statistical discrimination of quantum channels. Several paths have been taken for such this problem. However, there is much to be done in the avenue of optimizing channel discrimination using classical codes. At least two open questions can be pointed out: how to construct low complexity encoding schemes that are interesting for channel discrimination and, more importantly, how to develop capacity-achieving protocols. This paper aims at presenting a solution to these questions using polar codes. Firstly, we characterize the information rate and reliability parameter of the channels under polar encoding. We also show that the error probability of the scheme proposed decays exponentially with the square root of the code length. Secondly, an analysis of the optimal quantum states to be used as probes is given. [ABSTRACT FROM AUTHOR]

A common method for reading information stored in DNA molecules is shotgun sequencing. This method outputs a histogram of the frequencies of all the molecules’ substrings of a given length $\ell $. To protect against noisy readings, the rank-modulation scheme encodes the information in the relative ranking of the substring frequencies, instead of their absolute values. However, the best rank-modulation codes for shotgun sequencing have low rates which are asymptotically vanishing. In this paper we propose new constructions of rank-modulation codes for shotgun sequencing. The first code construction is systematic, allowing the user to arbitrarily set the frequencies of a large subset of the substrings, which the encoder then completes to a permutation that may be realized by a DNA molecule. The construction is then improved by allowing the user to set the frequencies of additional substrings, at the cost of imposing constraints on the frequencies. The resulting codes have higher, non-vanishing rates, compared with previously known codes. As an example, for histograms of substrings of length $\ell =2$ , and an alphabet of size 4 (as in DNA molecules), we are able to construct a code with rate $\approx 0.909$ , whereas previously, the best construction resulted in a code with rate $\approx 0.654$. Additionally, the encoded information in our construction may be written to shorter DNA molecules than possible before. We also prove that the systematic codes constructed in this paper are the largest possible among all systematic codes. [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

Consider a massive random access scenario in which a small set of $k$ active users out of a large number of $n$ potential users need to be scheduled in $b\ge k$ slots. What is the minimum common feedback to the users needed to ensure that scheduling is collision-free? Instead of a naive scheme of listing the indices of the $k$ active users in the order in which they should transmit, at a cost of $k\log (n)$ bits, this paper shows that for the case of $b=k$ , the rate of the minimum fixed-length common feedback code scales only as $k \log (e)$ bits, plus an additive term that scales in $n$ as $\Theta (\log \log (n))$ for fixed $k$. If a variable-length code can be used, assuming uniform activity among the users, the minimum average common feedback rate still requires $k \log (e)$ bits, but the dependence on $n$ can be reduced to $O(1)$. When $b>k$ , the number of feedback bits needed for collision-free scheduling can be significantly further reduced. Moreover, a similar scaling on the minimum feedback rate is derived for the case of scheduling $m$ users per slot, when $k \le mb$. The problem of constructing a minimum collision-free feedback scheduling code is connected to that of constructing a perfect hashing family, which allows practical feedback scheduling codes to be constructed from perfect hashing algorithms. [ABSTRACT FROM AUTHOR]

The growing number of Internet of Thing (IoT) and Ultra-Reliable Low Latency Communications (URLCC) use cases in next generation communication networks calls for the development of efficient Forward Error Correction (FEC) mechanisms. These use cases usually imply using short to mid-sized information blocks and requires low-complexity and/or fast decoding procedures. This paper investigates the joint learning of short to mid block-length coding schemes and associated Belief-Propagation (BP) like decoders using Machine Learning (ML) techniques. An interpretable auto-encoder (AE) architecture is proposed, ensuring scalability to block sizes currently challenging for ML-based linear block code design approaches. By optimizing a coding scheme w.r.t. the targeted decoder, the proposed system offers a good complexity/performance trade-off compared to various codes from literature with length up to 128 bits. [ABSTRACT FROM AUTHOR]

This paper studies low-latency streaming codes for the multi-hop network. The source transmits a sequence of messages to a destination through a chain of relays, and requires the destination to reconstruct each message by its deadline. We assume that each communication link is subjected to a certain maximum number of packet erasures. The case of a single relay (a three-node network) was considered in Fong et al. (2020). A coding scheme known as symbol-wise decode and forward was proposed. In the present work, we propose an alternative scheme that is different from Fong et al. (2020) and still achieves the same rate as in Fong et al. (2020) for the one hop case as the field-size goes to infinity. Furthermore, our proposed scheme naturally generalizes to the case of multiple-relay nodes yielding new achievable rates for this setting. The main difference with Fong et al. (2020) is that our proposed scheme exploits the ability of the relay nodes to adapt the transmission based on the erasures on the previous link. Hence, we refer to our scheme as “state-dependent” and contrast it with the scheme in Fong et al. (2020) that is state-independent. Our scheme requires the relay nodes to append a header to the transmitted packets, and we show that the size of the header does not depend on the field-size of the code. We also derive an upper bound on the maximal streaming rate achievable over a network with an arbitrary number of relays. We show that this upper bound matches our achievable rate in the special case when the maximal number of erasures on the first link is greater than or equal to the maximal number of erasures on each of the following links, and the field size goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we consider variable-length coding over the memoryless binary asymmetric channel (BAC) with full noiseless feedback, including the binary symmetric channel (BSC) as a special case. In 2012, Naghshvar et al. introduced a coding scheme, which we refer to as the small-enough-difference (SED) coding scheme. For symmetric binary-input channels, the deterministic variable-length feedback (VLF) code constructed with the SED coding scheme asymptotically achieves both capacity and Burnashev’s optimal error exponent. Building on the work of Naghshvar et al., this paper extends the SED coding scheme to the BAC and develops a non-asymptotic VLF achievability bound that is shown to achieve both capacity and the optimal error exponent. For the specific case of the BSC, we develop an additional non-asymptotic VLF achievability bound using a two-phase analysis that leverages both a submartingale synthesis and a Markov chain time of first passage analysis. Numerical evaluations show that both new VLF achievability bounds outperform Polyanskiy’s achievability bound for variable-length stop-feedback codes. [ABSTRACT FROM AUTHOR]

Timing-based side and covert channels in processor caches continue to be a threat to modern computers. This work shows for the first time, a systematic, large-scale analysis of Arm devices and the detailed results of attacks the processors are vulnerable to. Compared to x86, Arm uses different architectures, microarchitectural implementations, cache replacement policies, etc., which affects how attacks can be launched, and how security testing for the vulnerabilities should be done. To evaluate security, this paper presents security benchmarks specifically developed for testing Arm processors and their caches. The benchmarks are evaluated with sensitivity tests, which examine how sensitive the benchmarks are to having a correct configuration in the testing phase. Further, to evaluate a large number of devices, this work leverages a novel approach of using a cloud-based Arm device testbed for architectural and security research on timing channels and runs the benchmarks on 34 different physical devices. In parallel, there has been much interest in secure caches to defend the various attacks. Consequently, this paper also investigates secure cache architectures using proposed benchmarks. Especially, this paper implements and evaluates secure PL and RF caches, showing the security of PL and RF caches, but also uncovers new weaknesses. [ABSTRACT FROM AUTHOR]

To further mitigate the BER error floor of $M$ -ary differential chaos shift keying (MDCSK) modulation for impulsive noise, a low-cost high-reliability coded modulation scheme without equalizations is desirable for impulsive noise. In this paper, a protograph-based low-density parity-check coded MDCSK-based bit-interleaved coded modulation (MDCSK-BICM) scheme is proposed for the scenario with impulsive noise obeying Bernoulli-Laplace distribution. Considering the Gaussian distribution of protograph extrinsic information transfer (PEXIT) and the benchmark of coding design, the U-shaped non-coherent capacity is derived with optimal code rate for square MDCSK, showing that the code rates are stable with $E_b/N_0$ increasing but still much lower than that of DS-16DQAM at code rate $>\!0.145$. The derived joint probability density function can be considered as approximately a Gaussian distribution, which is used to improve the impulsive noise fitted PEXIT. Finally, a new code under a novel principle is designed for better bit error rate performance based on the improved PEXIT analysis. Both PEXIT analysis and simulation results demonstrate that the proposed scheme achieves a better BER than that with a traditional scheme, and suggesting a research approach based on simple methods. This basic research work provides a guide to establish a framework and optimize the performance of transmission systems over practical power line communication. [ABSTRACT FROM AUTHOR]

Electromagnetic Transient (EMT) simulation tools are typically developed using conventional procedural programming languages. On the other hand, modern high-level and equation-based programming languages, such as Modelica, are currently available. Modelica allows formulating models that are easy to develop, maintain and understand by expressing what needs to be computed without stating how it should be computed. This paper presents a Modelica-based simulator for electromagnetic transients. It is demonstrated that this approach offers significant advantages for developing sophisticated models. Computational performance and accuracy are compared to a conventional EMT-type simulation tool. [ABSTRACT FROM AUTHOR]

In 1991, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei’s duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei’s duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result of this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of $\mathbb {Z}$ , from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for $w$ -demi-matroids defined over finite sets and $w$ -demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics. [ABSTRACT FROM AUTHOR]

For fixed integers $n$ and $b\geq k$ , let $A(b,k,n)$ the largest size of a subset of $\{1,2,\ldots,b\}^{n}$ such that, for any $k$ distinct elements in the set, there is a coordinate where they all differ. Bounding $A(b,k,n)$ is a problem of relevant interest in information theory and computer science, relating to the zero-error capacity with list decoding and to the study of $(b, k)$ -hash families of functions. It is known that, for fixed $b$ and $k$ , $A(b,k,n)$ grows exponentially in $n$. In this paper, we determine new exponential upper bounds for different values of $b$ and $k$. A first bound on $A(b,k,n)$ for general $b$ and $k$ was derived by Fredman and Komlós in the ’80s and improved for certain $b\neq k$ by Körner and Marton and by Arikan. Only very recently better bounds were derived for general $b$ and $k$ by Guruswami and Riazanov, while stronger results for small values of $b=k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan, and by Costa and Dalai. In this paper, we strengthen the bounds for some specific values of $b$ and $k$. Our contribution is a new computational method for obtaining upper bounds on the values of a quadratic form defined over discrete probability distributions in arbitrary dimensions, which emerged as a central ingredient in recent works. The proposed method reduces an infinite-dimensional problem to a finite one, which we manage to further simplify by means of a series of optimality conditions. [ABSTRACT FROM AUTHOR]

The family of Goppa codes is one of the most interesting subclasses of linear codes. As the McEliece cryptosystem often chooses a random Goppa code as its key, knowledge of the number of inequivalent Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. In this paper we present a new approach to give an upper bound on the number of inequivalent extended irreducible binary Goppa codes. To be more specific, let $n>3$ be an odd prime number and $q=2^{n}$ ; let $r\geq 3$ be a positive integer satisfying $\gcd (r,n)=1$ and $\gcd \big (r,q(q^{2}-1)\big)=1$. We obtain an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree $r$. [ABSTRACT FROM AUTHOR]

Point cloud is a set of three-dimensional points in arbitrary order, which is a popular representation of 3D scene in autonomous navigation and immersive applications in recent years. Compression becomes an inevitable issue due to the huge data volume of point cloud. In order to effectively compress attributes of those points, proper reordering is important. The existing voxel-based point cloud attributes compression scheme uses a naive scan for points reordering. In this paper, we theoretically analyzed 3C properties of point cloud, i.e., Compactness, Clustering and Correlation, of different scan-orders defined by different space filling curves and disclosed that the Hilbert curve can provide the best spatial correlation preservation compared with Z-order and Gray-coded curves. It is also statistically verified that the Hilbert curve always has the best ability of attributes correlation preservation for point clouds with different sparsity. We also proposed a fast and iterative Hilbert address code generation method to implement points reordering. The Hilbert scan-order could be combined with various point cloud attribute coding methods. Experiments show that the correlation preservation feature of the proposed scan-order can bring us 6.1% and 6.5% coding gain for prediction and transform coding, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we present a class of hybrid coupled serially concatenated codes (HC-SCC), in which coupling is achieved by re-encoding partial outputs from both the outer and the inner encoders. Firstly, we derive the exact density evolution (DE) equations over the binary erasure channels (BECs) for the HC-SCCs, which can be used to analyze the impact of the coupling ratios and the coupling memories on the performances of the HC-SCCs. Secondly, we present a genie-aided lower bound for the proposed HC-SCCs to estimate their error-floors. Thirdly, we use the DE equations to design randomly punctured HC-SCCs, including rate-compatible HC-SCCs. The analytical and the numerical results show that: 1) for comparable coupling memories, the iterative decoding thresholds of the proposed HC-SCC ensembles are better than those of the recently introduced spatially coupled serially concatenated code (SC-SCC) ensembles and the partially parity-coupled turbo code (PPC-TC) ensembles; 2) with outer coupling memory five and inner coupling memory five, the iterative decoding thresholds of the rate-compatible HC-SCCs over the BECs are 0.0003 away from the corresponding channel capacities for a wide range of coding rates; 3) simulation results over the additive white Gaussian noise (AWGN) channels and the BECs are provided to confirm the performance advantages of the HC-SCCs. Particularly, the simulation results show that the HC-SCCs admit lower error-floor than the SC-SCCs and the rate-compatible HC-SCCs perform better than the PPC-TCs, which are consistent with the results of DE analysis. [ABSTRACT FROM AUTHOR]

The strong secrecy transmission problem of the arbitrarily varying wiretap channel (AVWC) with input and state constraints is investigated in this paper. First, a stochastic-encoder code lower bound of the strong secrecy capacity is established by applying the type argument and Csiszár’s almost independent coloring lemma. Then, a superposition stochastic-encoder code lower bound of the secrecy capacity is provided. The superposition stochastic-encoder code lower bound can be larger than the ordinary stochastic-encoder code lower bound. Random code lower and upper bounds of the secrecy capacity of the AVWC with constraints are further provided. Based on these results, we further consider a special case of the model, namely severely less noisy AVWC, and give the stochastic-encoder code and random code capacities. It is proved that the stochastic-encoder code capacity of the AVWC with constraints is either equal to or strictly smaller than the corresponding random code capacity, which is consistent with the property of the ordinary AVC. Finally, some numerical examples are presented to better illustrate our capacity results. Compared to the soft covering lemma that requires the codewords to be generated i.i.d., our method has more relaxed requirements regarding codebooks. It is proved that the good codebooks for secure transmission can be generated by choosing codewords randomly from a given type set, which is critical when considering the AVWC with constraints. [ABSTRACT FROM AUTHOR]

A block code is called a locally recoverable code (LRC for short) with $(r,\delta)$ -locality, if subject to any $\delta -1$ erasure failures, every symbol in the encoding can still be recovered by accessing at most $r$ other symbols. Recently, it was discovered by several authors that a $q$ -ary optimal $(r,\delta)$ -LRC, i.e., an LRC achieving the generalized Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the classical $q$ -ary MDS codes where it is conjectured that the code length is upper bounded by $q+1$ (or $q+2$ for some special cases). In this paper, we further investigate constructions of optimal $(r,\delta)$ -LRCs along the line of using parity-check matrices. Inspired by classical Reed-Solomon codes and the work in Jin (2019), we equip parity-check matrices with the Vandermonde structure. It turns out that a parity-check matrix with the Vandermonde structure that produces an optimal LRC must obey certain disjoint property for subsets of $\mathbb {F}_{q}$. We manage to show the existence of these disjoint subsets. Thus, this yields an optimal $(r,\delta)$ -LRC with code length $\Omega \left({q^{\frac {\delta }{2}\left({1+\frac {1}{\lfloor \frac {d-1}{\delta }\rfloor -1}}\right)}}\right)\vphantom {\bigg)_{j}}$ , where $d$ is the minimum distance. In particular, to our surprise, for $\delta =2$ this disjoint condition is equivalent to a well-studied problem in extremal graph theory. With the help of extremal graph theory, we succeed to improve all of the best known results in Guruswami et al. (2018) for $d\geq 7$. In addition, for $d=6$ , we are able to remove the constraint required in Jin (2019) that $q$ is even. [ABSTRACT FROM AUTHOR]

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [ABSTRACT FROM AUTHOR]

Erasure codes are essential for providing efficient resilience against node failures in distributed storage. Typically, an $[n, k]$ erasure code encodes $k$ symbols into $n$ symbols which are then stored in different nodes. Recent work by Kadekodi et al. shows that the failure rates of storage nodes vary significantly over time, and that changing the rate of the code (via a change in $n$ and $k$) in response to such variations provides substantial storage space savings. However, the resource overhead of re-encoding the already encoded data is prohibitively high. We present a new theoretical framework formalizing code conversion—the process of converting data encoded with an $[n^{ I}, k^{ I}]$ code into data encoded with an $[{n^{ F}}, {k^{ F}}]$ code while maintaining desired decodability properties. We then introduce convertible codes, a new class of codes that allow for code conversions in a resource-efficient manner. This paper begins the study on convertible codes by focusing on linear MDS codes and the access cost of conversion. We derive a lower bound on the access cost of conversion and present an explicit optimal construction matching this bound for an important subclass of conversions. Additionally, we propose constructions with low field-size requirement for a broad subset of parameters. Our results show that it is possible to achieve code conversions with significantly less resources than the default approach of re-encoding for a wide range of parameters. [ABSTRACT FROM AUTHOR]

The Euclidean hull of a linear code $C$ is the intersection of $C$ with its Euclidean dual $C^\perp $. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented. [ABSTRACT FROM AUTHOR]

Sliding window decoding has been employed as the decoding scheme of quasi-cyclic spatially-coupled low-density parity-check (QC-SC-LDPC) codes, with the decoding latency and complexity independent of the coupling length but proportional to the window size. The window size was empirically chosen at least three times the constraint length in previous works, which results in the preference of QC-SC-LDPC codes with small constraint length, to reduce the decoding latency and complexity in practical systems. However, the code design freedom and the decoder throughput are limited for these QC-SC-LDPC codes. In this paper, the optimal window size under the constraint of practical decoding complexity is analyzed employing multi-edge-type density evolution (MET-DE). It can be verified from the MET-DE analytical result that the optimal window size could achieve less than twice the constraint length for several edge spreading patterns, which largely loose this restriction in practical code design. Furthermore, it could be observed that besides the constraint length, the specific structure of the edge spreading pattern could affect the optimal window size as well, and thus the MET-DE analytical method for the optimal window size could effectively guide the design of QC-SC-LDPC codes in practical systems. Then an improved sliding window decoding scheme is proposed, including the optimal choice of window size and a parity-check based early termination scheme. Finally, the performance and robustness of the proposed sliding window decoding scheme for QC-SC-LDPC codes are demonstrated via simulation, with the decoding complexity at most twice as that of the conventional LDPC coded schemes. [ABSTRACT FROM AUTHOR]

This paper presents an alternative approach to real-valued signal representation and proposes an improved successive interference cancellation (SIC)-based detection scheme using the alternative signal representation for non-uniform constellations (NUCs) in Advanced Television Systems Committee (ATSC) 3.0. Since I/Q polarization interleaving in ATSC 3.0 multiple-input multiple-output (MIMO) precoding performs a non-linear operation, it makes it difficult to use conventional suboptimal detection schemes based on interference cancellation. Therefore, the alternative signal representation is designed to easily apply various suboptimal detection schemes. Further, to obtain more robustness and lower complexity, a suboptimal detection scheme based on the alternative signal representation and interference cancellation with QR decomposition is proposed. The proposed detection scheme exploits the structural properties of NUCs and jointly detects the real and imaginary parts as a block. In addition, by considering one or more candidate symbols in the interference cancellation for the proposed detection scheme, the performance and the complexity are trade-offs according to the number of the considering candidate symbols. Simulation results show that the proposed detection scheme with a slight increase of the complexity significantly outperforms a linear detection under fixed and mobile channels. [ABSTRACT FROM AUTHOR]

Spinal codes are a new family of rateless codes which have been proved to be capacity-achieving over both the additive white Gaussian noise (AWGN) channel and the binary symmetric channel (BSC). However, over the Rayleigh fading channel between the vehicle nodes, the error performance of Spinal codes is not satisfactory since the tail message blocks of Spinal codes are prone to error. To solve this issue, this paper proposes a new practical partial self-concatenation coding structure of Spinal codes, named $N$ tail-protected Spinal codes, to pointedly protect tail message blocks of Spinal codes over the flat Rayleigh fading channel. Theoretical performance analysis and simulation results are also provided to validate and verify the effectiveness of the proposed $N$ tail-protected Spinal codes. [ABSTRACT FROM AUTHOR]

Matrix multiplication is a fundamental building block in various distributed computing algorithms. In order to multiply large matrices, it is common practice to distribute the computation into multiple tasks running on different nodes. In order to tolerate stragglers among such nodes, various coding schemes have been proposed by adding additional coded tasks. However, most existing coding schemes for matrix multiplication are constructed for only one matrix multiplication, while batch matrix multiplication is common in large-scale distributed computing workloads. In this paper, we propose Rook Coding (RC), a novel polynomial-based coding framework for computing the multiplication of $n$ pairs of matrices in batch. Designed to achieve lower encoding time in practice, we construct RC as polynomials of much simpler forms than existing coding schemes for batch matrix multiplication, achieving a recovery threshold of $O(n^{\log _{2} ~3})$. Compared to existing coding schemes, RC achieves a lower encoding complexity in practice, because of its simpler forms in the encoding polynomials. Through extensive experiments, we show that RC can save the time of the whole job thanks to its low overhead of encoding. [ABSTRACT FROM AUTHOR]

We study spatially coupled LDPC codes that allow access to sub-blocks much smaller than the full code block. Sub-block access is realized by a semi-global decoder that decodes a chosen target sub-block by only accessing the target, plus a prescribed number of helper sub-blocks adjacent in the code chain. This paper develops a theoretical methodology for analyzing the semi-global decoding performance of spatially coupled LDPC codes constructed from protographs. The main result shows that semi-global decoding thresholds can be derived from certain thresholds we define for the single-sub-block graph. These characterizing thresholds are also used for deriving lower bounds on the decoder’s performance over channels with variability across sub-blocks, which are motivated by applications in data storage. [ABSTRACT FROM AUTHOR]

Recently, subfield codes of geometric codes over large finite fields ${\mathrm {GF}}(q)$ with dimension 3 and 4 were studied and distance-optimal subfield codes over ${\mathrm {GF}}(p)$ were obtained, where $q=p^{m}$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of $[q+1, 2, q]$ MDS codes over ${\mathrm {GF}}(q)$ , and then studies the subfield codes over ${\mathrm {GF}}(p)$ of some of the $[q+1, 2,q]$ MDS codes over ${\mathrm {GF}}(q)$. Two families of dimension-optimal codes over ${\mathrm {GF}}(p)$ are obtained, and several families of nearly optimal codes over ${\mathrm {GF}}(p)$ are produced. Several open problems are also proposed in this paper. [ABSTRACT FROM AUTHOR]

We consider the concatenation of a convolutional code (CC) with an optimized cyclic redundancy check (CRC) code as a promising paradigm for good short blocklength codes. The resulting CRC-aided convolutional code naturally permits the use of serial list Viterbi decoding (SLVD) to achieve maximum-likelihood decoding. The convolutional encoder of interest is of rate- $1/\omega $ and the convolutional code is either zero-terminated (ZT) or tail-biting (TB). The resulting CRC-aided convolutional code is called a CRC-ZTCC or a CRC-TBCC. To design a good CRC-aided convolutional code, we propose the distance-spectrum optimal (DSO) CRC polynomial. A DSO CRC search algorithm for the TBCC is provided. Our analysis reveals that the complexity of SLVD is governed by the expected list rank which converges to 1 at high SNR. This allows a good performance to be achieved with a small increase in complexity. In this paper, we focus on transmitting 64 information bits with a rate-1/2 convolutional encoder. For a target error probability $10^{-4}$ , simulations show that the best CRC-ZTCC approaches the random-coding union (RCU) bound within 0.4 dB. Several CRC-TBCCs outperform the RCU bound at moderate SNR values. [ABSTRACT FROM AUTHOR]

Previous algorithms for decoding AG codes up to the designed distance all assumed existence of an extra rational place on the base algebraic curve. The place is used to solve the decoding problem by linear algebra over the base field of the curve. The rationality of the place is essential, and therefore AG codes supported by all rational places on the curve are excluded from the domain of applicability of the decoding algorithms. This paper presents a decoding algorithm for those AG codes using an extra place of higher degree. Hence finally all AG codes, as Goppa defined 40 years ago, are equipped with a fast decoding algorithm. [ABSTRACT FROM AUTHOR]

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$ -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$ -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes. [ABSTRACT FROM AUTHOR]

This paper proposes a new probabilistic amplitude shaping (PAS) approach for concatenated two-level multi-level coding (MLC). The proposed system is based on a concatenated forward error correction (FEC) scheme where outer codes are serially concatenated with inner two-level MLC. This concatenated two-level MLC scheme has recently been shown to have a potential for achieving better performance-complexity trade-offs than the conventional bit-interleaved coded modulation (BICM). Meanwhile, PAS has recently been demonstrated to offer remarkable performance gains as well as rate adaptivity. However, the majority of existing works on PAS assume the use of the binary reflected Gray code as a bit-labeling, and its application to coded modulation schemes with other bit-labelings, such as two-level MLC, may not be straightforward. In this paper, we devise a bit-labeling scheme and propose a new PAS structure for an efficient integration of PAS with two-level MLC systems. More specifically, we propose to generate signed amplitude symbols with the distribution matcher (DM) for maximizing both coding and shaping gains achieved by two-level MLC and PAS, respectively, while the conventional PAS generates unsigned amplitude symbols. It is demonstrated by simulation results that, with 256QAM and inner polar codes, the proposed two-level MLC with PAS simultaneously offers 75% reduction in the number of required inner encoding and soft-decision (SD) decoding operations for given outer and inner FEC code lengths, and up to 0.3 dB performance gain over the conventional PAS scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose an optimization method for protograph-based spatially coupled low-density parity-check (SC-LDPC) codes under window decoding (WD). Previous works on constructing SC-LDPC codes for WD typically focused on optimizing asymptotic performance metrics such as the WD threshold. However, in this paper, it is observed that the WD threshold is not an appropriate metric to sufficiently explain the finite-length behavior of SC-LDPC codes under WD. Thus, we propose a new performance metric, called the window mean parameter, based on a scaling analysis to capture the WD performance more accurately and formulate a code optimization algorithm that optimizes the proposed performance metric. Since the proposed metric depends on the window size, the optimization algorithm can provide a code family of SC-LDPC codes optimized for various target window sizes. Simulation results confirm that the improvement in the proposed metric leads to a finite-length performance improvement, resulting in one to two orders of the frame error rate gain over the conventional SC-LDPC codes for a wide range of window sizes. Furthermore, we investigate structural characteristics of the proposed codes to provide a supplementary explanation for the performance improvement, which also promotes a better understanding of SC-LDPC codes for WD. [ABSTRACT FROM AUTHOR]

Insertion–deletion codes (insdel codes for short) play an important role in synchronization error correction. The higher the minimum insdel distance, the more insdel errors the code can correct. Haeupler and Shahrasbi established the Singleton bound for insdel codes: the minimum insdel distance of any $[n,k]$ linear code over $\mathbb {F}_{q}$ satisfies $d\leq 2n-2k+2$. There have been some constructions of insdel codes through Reed-Solomon codes with high capabilities, but none has come close to this bound. Recently, Do Duc et al. showed that the minimum insdel distance of any $[n,k]$ Reed-Solomon code is no more than $2n-2k$ if $q$ is large enough compared to the code length $n$ ; optimal codes that meet the new bound were also constructed explicitly. The contribution of this paper is twofold. We first show that the minimum insdel distance of any $[n,k]$ linear code over $\mathbb {F}_{q}$ satisfies $d\leq 2n-2k$ if $n > k > 1$. This result improves and generalizes the previously known results in the literature. We then give a sufficient condition under which the minimum insdel distance of a 2-D Reed-Solomon code of length $n$ over $\mathbb {F}_{q}$ is exactly equal to $2n-4$. As a consequence, we show that the sufficient condition is not hard to achieve; we explicitly construct an infinite family of optimal 2-D Reed-Somolom codes meeting the bound. [ABSTRACT FROM AUTHOR]

Linear codes with few weights have applications in consumer electronics, communications, data storage systems, secret sharing, authentication codes, and association schemes. As a special class of linear codes, minimal linear codes have important applications in secret sharing and secure computation of data between two parties. The construction of minimal linear codes with new and desirable parameters is an interesting research topic in coding theory and cryptography. Recently, Mesnager et.al. stated that “constructing linear codes with good parameters from non-weakly regular bent functions is an interesting problem.” The goal of this paper is to construct linear codes with two or three weights from non-weakly regular bent functions over finite fields and analyze the minimality of the constructed linear codes. In doing so, we draw inspiration from a paper by Mesnager in which she constructed linear codes with small weights from weakly regular bent functions based on a generic construction method. First, we recall the definitions of the subsets $B_{+}(f)$ and $B_{-}(f)$ associated with a non-weakly regular bent function $f$. Next, we construct two- or three-weight linear $p$ -ary codes on these sets using duals of the non-weakly regular bent functions that are also bent. We note that the constructed linear codes are minimal in almost all cases. Moreover, when $f$ is a non-weakly regular bent function in a certain subclass of Generalized Maiorana-McFarland bent functions, we determine the weight distributions of the corresponding linear codes. As far as we know, the construction of linear codes from non-weakly regular bent functions over finite fields is first studied in the literature by the second author in his dissertation. [ABSTRACT FROM AUTHOR]

Sparse code multiple access (SCMA) is a promising non-orthogonal multiple access technique to support massive connectivity for future wireless Internet of Things (IoT) networks. As the main feature of SCMA, modulation and spread spectrum is embedded into codebook mapping, offering significant codebook shaping gains to mitigate inter-cell interference. Maximizing the constellation-constrained average mutual information (AMI) is an effective way for SCMA codebook optimization. However, deriving a closed-form expression of the AMI is analytically intractable, while it is computationally costly to estimate the AMI by numerical methods. To address this challenge, this paper first derives a lower bound of the AMI with a closed-form expression. On this basis, we propose a novel codebook optimization method referred to as joint bare bones particle swarm optimization (JBBPSO) through maximizing the AMI lower bound. The proposed low-complexity method jointly optimizes the mother codebook including basic constellation and other non-zero-dimensional constellations, and the rotation angles of multiple users. Numerical results show that our proposed optimized codebooks outperform the state-of-the-art SCMA codebooks in terms of both the lower bound and the error performance. [ABSTRACT FROM AUTHOR]

This paper presents a comparative performance and complexity analysis of the advanced ISDB-T and ATSC 3.0 Low-Density Parity-Check (LDPC) codes using the Sum-Product Algorithm and Min-Sum Algorithm LDPC decoders. The performance value for each of the code rates of the LDPC codes over an additive white Gaussian noise (AWGN) channel and Rayleigh fading channel is presented. The mathematical representation of each of the SPA and MSA decoders is presented along with the calculated and measured numerical complexity values. [ABSTRACT FROM AUTHOR]

The prefixed Varshamov-Tenengolts (VT) codes, which are subsets of VT codes with predetermined prefixes, can be used for error correction over segmented edit channels. In this paper, we investigate the construction and analysis of this class of codes. First, we derive upper bounds on the size of zero-error codes for segmented edit channels with segment-by-segment decoding. Second, we establish a one-to-one correspondence between prefixed VT codes and Levenshtein codes. Based on this relation, we can obtain explicit formulas on the size of prefixed VT codes via the existing results on the size of Levenshtein codes. Third, we construct a new zero-error prefixed VT code and show that the size of the constructed code is strictly larger than that of the existing prefixed VT code for the segmented deletion channel. Finally, an efficient systematic encoding method of prefixed VT codes is proposed for the segmented edit channels. [ABSTRACT FROM AUTHOR]

Conventional design methods of polar-coded modulation schemes aim to minimize the block error rate (BLER) under successive cancellation (SC) decoding. However, codes designed by conventional methods are not competitive under successive cancellation list (SCL) decoding. This paper presents a new design method based on the BLER upper bound under maximum-likelihood (ML) decoding (ML-BLER upper bound). The ML-BLER upper bound depends on the weight enumerating function (WEF) of the polar-coded modulation scheme over the squared Euclidean distance. In this paper, the polar-coded modulation is randomized by the concept of interleaved polar (i-polar) codes, and the WEF averaged over the ensemble of the polar-coded modulation schemes can be derived. Three polar-coded modulation schemes are considered, i.e., the bit-interleaved polar-coded modulation with a single interleaver (BIPCM-SI), the bit-interleaved polar-coded modulation with multiple interleavers (BIPCM-MI), and the multi-level polar-coded modulation (MLPCM). A new bit channel selection algorithm for polar-coded modulation schemes is proposed, which takes the polarization effect and the ML-BLER upper bound as design criteria. Design examples show that, under SCL decoding, the polar-coded modulation schemes (without CRC) with the proposed channel selection algorithm outperform those with conventional algorithms and are competitive as compared to the state-of-the-art 5G LDPC codes. [ABSTRACT FROM AUTHOR]

The researches of Golay complementary sequences (GCSs) over 16 and 64 quadrature amplitude modulation (QAM) from 2001 to 2008 were generalized to $4^{q} $ -QAM GCSs of length $2^{m}$ by Li (the generalized cases I-III for $q\ge 2$) in 2010 and Liu et al. (the generalized cases IV-V for $q\ge 3$) in 2013. Those sequences are presented by the combination of the quaternary standard GCSs and compatible offsets. By providing new compatible offsets based on the factorization of the integer $q$ , we propose two new constructions of $4^{q} $ -QAM GCSs, which have the generalized cases I-V as special cases. The numbers of the proposed GCSs are equal to the product of the number of the quaternary standard GCSs and the number of the compatible offsets. Denote the number of prime factors of $q$ counted with multiplicity by $\Omega (q)$. The number of new offsets in our first construction is lower bounded by a polynomial of $m$ with degree $\Omega (q)$ , while the numbers of offsets in the generalized cases I-III and IV-V are a linear polynomial and a quadratic polynomial of $m$ , respectively. If $q$ has a prime factor larger than 2, the number of new offsets in our second construction is lower bounded by a polynomial of $m$ with degree $\Omega (q)+1$. As an example, the new offsets in our two constructions for $q=6$ , whose number is bounded by a cubic polynomial, is also given. The proof in this paper implies that all the mentioned GCSs over QAM can be regarded as projections of Golay complementary arrays of size $2\times 2\times \cdots \times 2$. [ABSTRACT FROM AUTHOR]

It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper. [ABSTRACT FROM AUTHOR]