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We propose a successive cancellation list (SCL) encoding and decoding scheme for the Gelfand Pinsker (GP) problem based on the known nested polar coding scheme. It applies SCL encoding for the source coding part, and SCL decoding with a properly defined CRC for the channel coding part. The scheme shows improved performance compared to the existing method. A known issue with nested polar codes for binary dirty paper is the existence of frozen channel code bits that are not frozen in the source code. These bits need to be retransmitted in a second phase of the scheme, thus reducing the rate and increasing the required blocklength. We provide an improved bound on the size of this set, and on its scaling with respect to the blocklength, when the Bhattacharyya parameter of the test channel used for source coding is sufficiently large, or the Bhattacharyya parameter of the channel seen at the decoder is sufficiently small. The result is formulated for an arbitrary binary-input memoryless GP problem, since unlike the previous results, it does not require degradedness of the two channels mentioned above. Finally, we present simulation results for binary dirty paper and noisy write once memory codes. [ABSTRACT FROM AUTHOR]

How to model the effect of reflection is crucial for single image reflection removal (SIRR) task. Modern SIRR methods usually simplify the reflection formulation with the assumption of linear combination of a transmission layer and a reflection layer. However, the large variations in image content and the real-world picture-taking conditions often result in far more complex reflection. In this paper, we introduce a new screen-blur combination based on two important factors, namely the intensity and the blurriness of reflection, to better characterize the reflection formulation in SIRR. Specifically, we present Screen-blur Reflection Networks (SRNet), which executes the screen-blur formulation in its network design and adapts to the complex reflection on real scenes. Technically, SRNet consists of three components: a blended image generator, a reflection estimator and a reflection removal module. The image generator exploits the screen-blur combination to synthesize the training blended images. The reflection estimator learns the reflection layer and a blur degree that measures the level of blurriness for reflection. The reflection removal module further uses the blended image, blur degree and reflection layer to filter out the transmission layer in a cascaded manner. Superior results on three different SIRR methods are reported when generating the training data on the principle of the screen-blur combination. Moreover, extensive experiments on six datasets quantitatively and qualitatively demonstrate the efficacy of SRNet over the state-of-the-art methods. [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 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]

The binary quadratic-residue codes and the punctured Reed-Muller codes ${\mathcal {R}}_{2}((m-1)/2, m))$ are two families of binary cyclic codes with parameters $[n, (n+1)/2, d \geq \sqrt {n}]$. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length $2^{m}-1$ and dimension near $2^{m-1}$ with good minimum distances. When $m \geq 3$ is odd, the codes become a family of duadic codes with parameters $[2^{m}-1, 2^{m-1}, d]$ , where $d \geq 2^{(m-1)/2}+1$ if $m \equiv 3 \pmod {4}$ and $d \geq 2^{(m-1)/2}+3$ if $m \equiv 1 \pmod {4}$. The two families of binary cyclic codes contain some optimal binary cyclic codes. [ABSTRACT FROM AUTHOR]

Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing minimal binary codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity, are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension $m$ and length less than or equal to $m(m+1)/2$ are obtained. Besides, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields additionally to symmetric cryptography, such as coding theory and secret sharing schemes. [ABSTRACT FROM AUTHOR]

This paper evaluates the practically achievable best error performance obtained from a dimmable visible light communication (VLC) system that conveys information using binary modulations, such as on-off keying (OOK). As compared to previous studies that address the ideal performance of dimmable VLC systems only over reliable channels with arbitrarily small error probability, this paper presents the best achievable error probability of various configurations with the dimming targets, signal quality, and data rate. The numerical results establish the ultimate decoding error performance targeted for dimmable VLC systems and are expected to provide a practical guideline for the enhanced VLC system design. [ABSTRACT FROM AUTHOR]

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(ƒ) = {Tr(aƒ(x) + bx)x∈Fqm*: a, b ∈ Fqm}, where q is a prime power, Fqm* = Fqm \ {0}, Tr is the trace function from Fqm to Fq, and ƒ(x) is a function from Fqm to Fqm with ƒ(0) = 0. Almost bent functions, quadratic functions and some monomials on F2m were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we apply two-to-one functions over F2n in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) (x2t + x)e with gcd (t, n) = gcd(e, 2n−1) = 1. Based on the study of the Walsh transforms of those functions or their variants, we present many classes of linear codes with few nonzero weights, including one weight, three weights, four weights, and five weights. The weight distributions of the proposed codes with one weight and with three weights are determined. In addition, we discuss the minimum distance of the dual of the constructed codes and show that some of them achieve the sphere packing bound. Moreover, examples show that some codes in this paper have best-known parameters. [ABSTRACT FROM AUTHOR]

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure a Boolean function’s resistance to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. Some of the binary LCD codes presented in this paper are optimal. These binary LCD codes have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems. [ABSTRACT FROM AUTHOR]

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and Fq is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n−1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q. [ABSTRACT FROM AUTHOR]

$\mathbb {Z}_{2^{s}}$ -additive codes are subgroups of $\mathbb {Z}^{n}_{2^{s}}$ , and can be seen as a generalization of linear codes over $\mathbb {Z}_{2}$ and $\mathbb {Z}_{4}$. A $\mathbb {Z}_{2^{s}}$ -linear code is a binary code (not necessarily linear) which is the Gray map image of a $\mathbb {Z}_{2^{s}}$ -additive code. We consider $\mathbb {Z}_{2^{s}}$ -additive simplex codes of type $\alpha $ and $\beta $ , which are a generalization over $\mathbb {Z}_{2^{s}}$ of the binary simplex codes. These codes are related to the $\mathbb {Z}_{2^{s}}$ -additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding $\mathbb {Z}_{2^{s}}$ -linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly, $\mathbb {Z}_{2^{s}}$ -additive MacDonald codes are defined for $s>2$ , and equivalent results are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we determine all dihedral subgroups in $PGL_{2}(\mathbb {F}_{q})$ , where $q=2^{l}$ and $l$ is a positive integer; and construct binary expurgated and extended Goppa codes with dihedral automorphism groups. Moreover, it is easy to obtain binary quasi-cyclic expurgated or extended Goppa codes. [ABSTRACT FROM AUTHOR]

In this paper, we focus on decoding nonbinary low-density parity-check (LDPC) codes in Galois fields of characteristic two via the proximal alternating direction method of multipliers (proximal-ADMM). By exploiting Flanagan/Constant-Weighting embedding techniques and the decomposition technique based on three-variables parity-check equations, two efficient proximal-ADMM decoders for nonbinary LDPC codes are proposed. We show that both of them are theoretically guaranteed convergent to some stationary point of the decoding model and either of their computational complexities in each proximal-ADMM iteration scales linearly with LDPC code’s length and the size of the considered Galois field. Moreover, the decoder based on the Constant-Weight embedding technique satisfies the favorable property of codeword symmetry. Simulation results demonstrate their effectiveness in comparison with state-of-the-art LDPC decoders. [ABSTRACT FROM AUTHOR]

This paper takes a fresh look at the image authentication problem and proposes an alternative framework for personalized authentication based on the hash learning technology. Conventional image authentication methods tend to provide a general authentication framework for all images with the fixed quantization strategy and fixed control parameters determined based on given limited images and attacks. However, they may suffer from more or less misjudgments in practice, not to mention performance degradation when encountering out-of-sample images. Instead of proposing a new feature extraction algorithm, a novel personalized authentication framework which incorporates the distance metric learning technology and supervised quantization strategy to the process of image authentication is proposed in this paper. The tamper detection task is reformulated as a new supervised manipulation classification problem. For each input image, various content-preserving and content-changing samples are generated automatically firstly. Then, feature representations of all samples can be obtained by existing feature extraction methods. After that, a weighted large margin for manipulation classification (WLMMC) scheme is proposed to learn an effective feature mapping space to improve the classification performance between content-changing samples and content-preserving samples. During the quantization stage, a novel supervised personalized quantization strategy (SPQ), which is motivated by the observation that different attacks have different degrees of influence on feature components, is proposed to learn more compact yet discriminative binary codes for each input image. Effectiveness of the proposed framework is qualitatively and quantitatively demonstrated on a variety of images. Extensive experiments show that the proposed framework can significantly improve the authentication performance over the state-of-the-art techniques while achieve more compact hash codes flexibly as required. [ABSTRACT FROM AUTHOR]

In this paper, we present a variable-length binary code for permutations of degree n, where n is a power of 2. The Lehmer code and its variants provide a bijection between permutations and the indexes in lexicographic ordering. They provide compression of permutations approaching Shannon bound at the expense of structural information. The variable-length code proposed in this paper has asymptotically optimal compression capability while preserving structural information of permutations. In other words, the code encodes the way a permutation is organized, and is optimal in the sense that the ratio of average codeword length and the entropy of permutations approaches 1 as n tends to infinity. The encoding and decoding are efficient. Like the Lehmer code and other enumerative codes, it is not necessary to construct look-up tables. The code is complete and satisfies the prefix condition. Furthermore, the codeword length indicates how “well shuffled” a permutation is. Permutations with longer codewords seem more “random.” An enumeration method of the variable-length code is also given by using T. Cover’s enumerative coding scheme and Schalkwijk code. This gives a different indexing from the Lehmer code. This work can be extended to the case of arbitrary $n$ in a straightforward way. [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]

Non-overlapping codes are a set of codewords in $\bigcup _{n \ge 2} \mathbb {Z}_{q}^{n}$ , where $\mathbb {Z}_{q} = \{0,1, {\dots },q-1\}$ , such that the prefix of each codeword is not a suffix of any codeword in the set, including itself; and for variable-length codes, a codeword does not contain any other codeword as a subword. In this paper, we investigate a generic method to generalize binary codes to $q$ -ary ones for $q > 2$ , and analyze this generalization on the two constructions given by Levenshtein (also by Gilbert; Chee, Kiah, Purkayastha, and Wang) and Bilotta, respectively. The generalization on the former construction gives large non-expandable fixed-length non-overlapping codes whose size can be explicitly determined; the generalization on the latter construction is the first attempt to generate $q$ -ary variable-length non-overlapping codes. More importantly, this generic method allows us to utilize the generating function approach to analyze the cardinality of the underlying $q$ -ary non-overlapping codes. The generating function approach not only enables us to derive new results, e.g., recurrence relations on their cardinalities, new combinatorial interpretations for the constructions, and the superior limit of their cardinalities for some special cases, but also greatly simplifies the arguments for these results. Furthermore, we give an exact formula for the number of fixed-length words that do not contain the codewords in a variable-length non-overlapping code as subwords. This thereby solves an open problem by Bilotta and induces a recursive upper bound on the maximum size of variable-length non-overlapping codes. [ABSTRACT FROM AUTHOR]

In entropy coding, universal coding of integers (UCI) is a binary universal prefix code, such that the ratio of the expected codeword length to $\max \{1, H(P)\}$ is less than or equal to a constant expansion factor $K_{\mathcal {C}}$ for any probability distribution $P$ , where $H(P)$ is the Shannon entropy of $P$. $K_{\mathcal {C}}^{*}$ is the infimum of the set of expansion factors. The optimal UCI is defined as a class of UCI possessing the smallest $K_{\mathcal {C}}^{*}$. Based on prior research, the range of $K_{\mathcal {C}}^{*}$ for the optimal UCI is $2\leq K_{\mathcal {C}}^{*}\leq 2.75$. Currently, the code constructions achieve $K_{\mathcal {C}}=2.75$ for UCI and $K_{\mathcal {C}}=3.5$ for asymptotically optimal UCI. In this paper, we construct a class of UCI, termed $\iota $ code, to achieve $K_{\mathcal {C}}=2.5$. This further narrows the range of $K_{\mathcal {C}}^{*}$ to $2\leq K_{\mathcal {C}}^{*}\leq 2.5$. Next, a family of asymptotically optimal UCIs is presented, where their expansion factor infinitely approaches 2.5. Then, tighter upper bounds of the expansion factors are derived for several classic UCIs. Finally, we prove that the length of the first codeword of optimal UCI is one. [ABSTRACT FROM AUTHOR]

With the dramatic increase in the amount of multimedia data, cross-modal similarity retrieval has become one of the most popular yet challenging problems. Hashing offers a promising solution for large-scale cross-modal data searching by embedding the high-dimensional data into the low-dimensional similarity preserving Hamming space. However, most existing cross-modal hashing usually seeks a semantic representation shared by multiple modalities, which cannot fully preserve and fuse the discriminative modal-specific features and heterogeneous similarity for cross-modal similarity searching. In this paper, we propose a joint specifics and consistency hash learning method for cross-modal retrieval. Specifically, we introduce an asymmetric learning framework to fully exploit the label information for discriminative hash code learning, where 1) each individual modality can be better converted into a meaningful subspace with specific information, 2) multiple subspaces are semantically connected to capture consistent information, and 3) the integration complexity of different subspaces is overcome so that the learned collaborative binary codes can merge the specifics with consistency. Then, we introduce an alternatively iterative optimization to tackle the specifics and consistency hashing learning problem, making it scalable for large-scale cross-modal retrieval. Extensive experiments on five widely used benchmark databases clearly demonstrate the effectiveness and efficiency of our proposed method on both one-cross-one and one-cross-two retrieval tasks. [ABSTRACT FROM AUTHOR]

Fine-grained hashing is a new topic in the field of hashing-based retrieval and has not been well explored up to now. In this paper, we raise three key issues that fine-grained hashing should address simultaneously, i.e., fine-grained feature extraction, feature refinement as well as a well-designed loss function. In order to address these issues, we propose a novel Fine-graIned haSHing method with a double-filtering mechanism and a proxy-based loss function, FISH for short. Specifically, the double-filtering mechanism consists of two modules, i.e., Space Filtering module and Feature Filtering module, which address the fine-grained feature extraction and feature refinement issues, respectively. Thereinto, the Space Filtering module is designed to highlight the critical regions in images and help the model to capture more subtle and discriminative details; the Feature Filtering module is the key of FISH and aims to further refine extracted features by supervised re- weighting and enhancing. Moreover, the proxy-based loss is adopted to train the model by preserving similarity relationships between data instances and proxy-vectors of each class rather than other data instances, further making FISH much efficient and effective. Experimental results demonstrate that FISH achieves much better retrieval performance compared with state-of-the-art fine-grained hashing methods, and converges very fast. The source code is publicly available: https://github.com/chenzhenduo/FISH. [ABSTRACT FROM AUTHOR]

Recommendation efficiency and data sparsity problems have been regarded as two challenges of improving performance for online recommendation. Most of the previous related work focus on improving recommendation accuracy instead of efficiency. In this paper, we propose a Deep Pairwise Hashing (DPH) to map users and items to binary vectors in Hamming space, where a user's preference for an item can be efficiently calculated by Hamming distance, which significantly improves the efficiency of online recommendation. To alleviate data sparsity and cold-start problems, the user-item interactive information and item content information are unified to learn effective representations of items and users. Specifically, we first pre-train robust item representation from item content data by a Denoising Auto-encoder instead of other deterministic deep learning frameworks; then we finetune the entire framework by adding a pairwise loss objective with discrete constraints; moreover, DPH aims to minimize a pairwise ranking loss that is consistent with the ultimate goal of recommendation. Finally, we adopt the alternating optimization method to optimize the proposed model with discrete constraints. Extensive experiments on three different datasets show that DPH can significantly advance the state-of-the-art frameworks regarding data sparsity and item cold-start recommendation. [ABSTRACT FROM AUTHOR]

The probability Jaccard similarity was recently proposed as a natural generalization of the Jaccard similarity to measure the proximity of sets whose elements are associated with relative frequencies or probabilities. In combination with a hash algorithm that maps those weighted sets to compact signatures which allow fast estimation of pairwise similarities, it constitutes a valuable method for big data applications such as near-duplicate detection, nearest neighbor search, or clustering. This paper introduces a class of one-pass locality-sensitive hash algorithms that are orders of magnitude faster than the original approach. The performance gain is achieved by calculating signature components not independently, but collectively. Four different algorithms are proposed based on this idea. Two of them are statistically equivalent to the original approach and can be used as drop-in replacements. The other two may even improve the estimation error by introducing statistical dependence between signature components. Moreover, the presented techniques can be specialized for the conventional Jaccard similarity, resulting in highly efficient algorithms that outperform traditional minwise hashing and that are able to compete with the state of the art. [ABSTRACT FROM AUTHOR]

Linear codes over $\mathbb {Z}_{p^{s}}$ of length $n$ are subgroups of $\mathbb {Z}_{p^{s}}^{n}$. These codes are also called $\mathbb {Z}_{p^{s}}$ -additive codes and can be seen as a generalization of linear codes over $\mathbb {Z}_{2}$ and $\mathbb {Z}_{4}$. A $\mathbb {Z}_{p^{s}}$ -linear code is a code over $\mathbb {Z}_{p}$ , not necessarily linear, which is the generalized Gray map image of a $\mathbb {Z}_{p^{s}}$ -additive code. In 2015, a systematic encoding was found for $\mathbb {Z}_{4}$ -linear codes. Moreover, an alternative permutation decoding method, which is suitable for any binary code (not necessarily linear) with a systematic encoding, was established. In this paper, we generalize these results by presenting a systematic encoding for any $\mathbb {Z}_{p^{s}}$ -linear code with $s\geq 2$ and $p$ prime. We also describe a permutation decoding method for any systematic code over $\mathbb {Z}_{p}$ , not necessarily linear, and show some examples of how to use this systematic encoding in this decoding method. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of $n \times n$ arrays affected by ${t}$ -criss-cross deletions defined as any combination of ${t_{\mathrm {r}}}$ row and ${t_{\mathrm {c}}}$ column deletions such that ${t_{\mathrm {r}}}+ {t_{\mathrm {c}}}= {t}$ for a given $t$. We show an equivalence between correcting ${t}$ -criss-cross deletions and ${t}$ -criss-cross insertions and show that a code correcting ${t}$ -criss-cross insertions/deletions has redundancy at least ${t} n + {t}\log n - \log ({t}!)$. Then, we present an existential construction of a ${t}$ -criss-cross insertion/deletion correcting code with redundancy bounded from above by ${t} n + \mathcal {O}({t}^{2} \log ^{2} n)$. The main ingredients of the presented code construction are systematic binary ${t}$ -deletion correcting codes and Gabidulin codes. The first ingredient helps locating the indices of the inserted/deleted rows and columns, thus transforming the insertion/deletion-correction problem into a row/column erasure-correction problem which is then solved using the second ingredient. [ABSTRACT FROM AUTHOR]

Self-supervised representation learning for videos has been very attractive recently because these methods exploit the information inherently obtained from the video itself instead of annotated labels that is quite time-consuming. However, existing methods ignore the importance of global observation while performing spatio-temporal transformation perception, which highly limits the expression capabilities of the video representation. This paper proposes a novel pretext task that combines the temporal information perception of the video with the motion amplitude perception of moving objects to learn the spatio-temporal representation of the video. Specifically, given a video clip containing several video segments, each video segment is sampled by different sampling rates and the order of video segments is disrupted. Then, the network is used to regress the sampling rate of each video segment and classify the order of input video segments. In the pre-training stage, the network can learn rich spatio-temporal semantic information where content-related contrastive learning is introduced to make the learned video representation more discriminative. To alleviate the appearance dependency caused by contrastive learning, we design a novel and robust vector similarity measurement approach, which can take feature alignment into consideration. Moreover, a view synthesis framework is proposed to further improve the performance of contrastive learning by automatically generating reasonable transformed views. We conduct benchmark experiments with several 3D backbone networks on two datasets. The results show that our proposed method outperforms the existing state-of-the-art methods across the three backbones on two downstream tasks of human action recognition and video retrieval. [ABSTRACT FROM AUTHOR]

Online image hashing has received increasing research attention recently, which processes large-scale data in a streaming fashion to update the hash functions on-the-fly. To this end, most existing works exploit this problem under a supervised setting, i.e., using class labels to boost the hashing performance, which suffers from the defects in both adaptivity and efficiency: First, large amounts of training batches are required to learn up-to-date hash functions, which leads to poor online adaptivity. Second, the training is time-consuming, which contradicts with the core need of online learning. In this paper, a novel supervised online hashing scheme, termed Fast Class-wise Updating for Online Hashing (FCOH), is proposed to address the above two challenges by introducing a novel and efficient inner product operation. To achieve fast online adaptivity, a class-wise updating method is developed to decompose the binary code learning and alternatively renew the hash functions in a class-wise fashion, which well addresses the burden on large amounts of training batches. Quantitatively, such a decomposition further leads to at least 75 percent storage saving. To further achieve online efficiency, we propose a semi-relaxation optimization, which accelerates the online training by treating different binary constraints independently. Without additional constraints and variables, the time complexity is significantly reduced. Such a scheme is also quantitatively shown to well preserve past information during updating hashing functions. We have quantitatively demonstrated that the collective effort of class-wise updating and semi-relaxation optimization provides a superior performance comparing to various state-of-the-art methods, which is verified through extensive experiments on three widely-used datasets. [ABSTRACT FROM AUTHOR]

A class of network codes have been proposed in the literature where the symbols transmitted on network edges are binary vectors and the coding operation performed in network nodes consists of the application of (possibly several) permutations on each incoming vector and XOR-ing the results to obtain the outgoing vector. These network codes, which we will refer to as permute-and-add network codes, involve simpler operations and are known to provide lower complexity solutions than scalar linear network codes. The complexity of these codes is determined by their degree which is the number of permutations applied on each incoming vector to compute an outgoing vector. Constructions of permute-and-add network codes for multicast networks are known. In this paper, we provide a new framework based on group algebras to design permute-and-add network codes for arbitrary (not necessarily multicast) networks. Our framework allows the use of any finite group of permutations (including circular shifts, proposed in prior works) and admits a trade-off between coding rate and the degree of the code. Further, our technique permits elegant recovery and generalizations of the key results on permute-and-add network codes known in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and defining sets. Second, we construct quaternary linear codes via simplicial complexes and determine the weight distributions of these codes. Third, the weight distributions of the binary subfield codes of these quaternary codes are also computed by employing the general characterization. Furthermore, we present two infinite families of optimal linear codes with respect to the Griesmer Bound, and a class of binary almost optimal codes with respect to the Sphere Packing Bound. We also need to emphasize that we obtain at least 9 new quaternary linear codes. [ABSTRACT FROM AUTHOR]

Neural Network Coding and Representation (NNR) is the first international standard for efficient compression of neural networks (NNs). The standard is designed as a toolbox of compression methods, which can be used to create coding pipelines. It can be either used as an independent coding framework (with its own bitstream format) or together with external neural network formats and frameworks. For providing the highest degree of flexibility, the network compression methods operate per parameter tensor in order to always ensure proper decoding, even if no structure information is provided. The NNR standard contains compression-efficient quantization and deep context-adaptive binary arithmetic coding (DeepCABAC) as core encoding and decoding technologies, as well as neural network parameter pre-processing methods like sparsification, pruning, low-rank decomposition, unification, local scaling and batch norm folding. NNR achieves a compression efficiency of more than 97% for transparent coding cases, i.e. without degrading classification quality, such as top-1 or top-5 accuracies. This paper provides an overview of the technical features and characteristics of NNR. [ABSTRACT FROM AUTHOR]

Approximate Nearest Neighbor(ANN) search is the core problem in many large-scale machine learning and computer vision applications such as multimodal retrieval. Hashing is becoming increasingly popular, since it can provide efficient similarity search and compact data representations suitable for handling such large-scale ANN search problems. Most hashing algorithms concentrate on learning more effective projection functions. However, the accuracy loss in the quantization step has been ignored and barely studied. In this paper, we analyse the importance of various projected dimensions, distribute them into several groups and quantize them with two types of values which can both better preserve the neighborhood structure among data. One is Variable Integer-based Quantization (VIQ) that quantizes each projected dimension with integer values. The other is Variable Codebook-based Quantization (VCQ) that quantizes each projected dimension with corresponding codebook values. We conduct experiments on five common public data sets containing up to one million vectors. The results show that the proposed VCQ and VIQ algorithms can both achieve much higher accuracy than state-of-the-art quantization methods. Furthermore, although VCQ performs better than VIQ, ANN search with VIQ provides much higher search efficiency. [ABSTRACT FROM AUTHOR]

This paper develops a non-coherent index modulation (IM) system in which activation patterns are characterized by multi-level block codes. We analyze performance of such a system under the maximum-likelihood (ML) receiver and when the set of activation patterns follows a multi-level code generated from asymptotically optimal alphabets. An asymptotic analysis of the pair-wise error probability (PEP) shows that the system can exploit a diversity order that is determined by the distance of the worst codeword pair in the $l_{1}$ metric, known as the Manhattan norm. We then explore the rate-diversity tradeoff for the developed non-coherent IM system as a function of the code length. Specifically, Gilbert-style bounds on the data rates for systems based on binary and ternary codes are obtained that can ensure a given diversity order. We approach the problem of packing in the $l_{1}$ metric by partitioning codes into permutation modulation codes (PMCs) and obtaining Gilbert-style bounds on PMCs. Several achievable rates for non-coherent binary and ternary IM systems, as well as a tradeoff between the information rate and codeword error probability (CEP) are also derived. Finally, simulation results are provided to corroborate the theoretical analysis. [ABSTRACT FROM AUTHOR]

The nonlocal self-similarity in natural image provides an effective prior for single image super-resolution (SISR), which is beneficial to contextual information capture and performance improvement, as demonstrated by conventional SISR methods. However, it is little explored to utilize this property in deep neural networks. In this paper, we propose a self-similarity prior guided (SSPG) network to incorporate self-similarity-based nonlocal operation into deep neural network for SISR. Specifically, we design a cross-scale nearest-neighbor residual (CSNNR) block via introducing cross-scale $k$ -nearest neighbors (KNN) matching into a residual block, which can be flexibly integrated into deep networks to capture long-range correlations among multi-scale and multi-level features. Meanwhile, by stacking a CSNNR block and a sequence of wide-activated residual blocks with a local skip-connection, a multi-level residual self-similarity (MRSS) module is developed to effectively employ local and nonlocal information for detail recovery. Thus, through cascading multiple MRSS modules, the proposed SSPG network performs both self-similarity-based nonlocal operation and convolution-based local operation on multi-level features to reconstruct informative features for accurate SISR. In addition, for pursuing visually pleasing results, we apply our SSPG network to the perception-oriented SISR field by following the framework of generative adversarial networks. In particular, we explore a sample-discriminating learning mechanism based on the statistical descriptions of training samples, and include it in optimization procedure to automatically tune the contributions of different samples according to their characteristics and then focus the network on creating realistic results. Extensive quantitative and qualitative evaluations on benchmark datasets illustrate the superiority of our proposed models over the state-of-the-art methods for both distortion-oriented and perception-oriented image super-resolution tasks. [ABSTRACT FROM AUTHOR]

In any transmission scheme, security against eavesdroppers is of importance. Contrary to classical encryption, security directly at the physical layer can be achieved by applying suited coding and modulation schemes. In this paper, coded modulation schemes, i.e., the use of higher-order constellations, is studied for generating both security against a wiretapper and reliability for the legitimate user. Specifically, a multilevel coding (MLC) scheme is considered. The influence of the constellation, in particular its cardinality and its dimensionality, on the gained security is assessed. As LSB component code, a practical secure coding scheme based on punctured LDPC codes is used. The security level for the employed LDPC code ensembles within MLC constructions is examined with an adapted density evolution technique. Results from numerical simulations cover the theoretical considerations and match with the derived asymptotic behavior. The results are compared to the situation when straightforwardly applying a binary code according to the BICM approach. [ABSTRACT FROM AUTHOR]

Some nonlinear codes, such as Kerdock and Preparata codes, can be represented as binary images under the Gray map of linear codes over rings. This paper introduces MAP decoding of Kerdock and Preparata codes by working with their quaternary representation (linear codes over $\mathbb {Z}_{4}$) with the complexity of $\mathcal {O}(N^{2}\log _{2} N)$ , where N is the code length in $\mathbb {Z}_{4}$. A sub-optimal bitwise APP decoder with good error-correcting performance and complexity of $\mathcal {O}(N\log _{2} N)$ that is constructed using the decoder lifting technique is also introduced. This APP decoder extends upon the original lifting decoder by working with likelihoods instead of hard decisions and is not limited to Kerdock and Preparata code families. Simulations show that our novel decoders significantly outperform several popular decoders in terms of error rate. [ABSTRACT FROM AUTHOR]

Due to the exponential growth of multimedia data, multi-modal hashing as a promising technique to make cross-view retrieval scalable is attracting more and more attention. However, most of the existing multi-modal hashing methods either divide the learning process unnaturally into two separate stages or treat the discrete optimization problem simplistically as a continuous one, which leads to suboptimal results. Recently, a few discrete multi-modal hashing methods that try to address such issues have emerged, but they still ignore several important discrete constraints (such as the balance and decorrelation of hash bits). In this paper, we overcome those limitations by proposing a novel method named “Enhanced Discrete Multi-modal Hashing (EDMH)” which learns binary codes and hashing functions simultaneously from the pairwise similarity matrix of data, under the aforementioned discrete constraints. Although the model of EDMH looks a lot more complex than the other models for multi-modal hashing, we are actually able to develop a fast iterative learning algorithm for it, since the subproblems of its optimization all have closed-form solutions after introducing a couple of auxiliary variables. Our experimental results on three real-world datasets have revealed the usefulness of those previously ignored discrete constraints and demonstrated that EDMH not only performs much better than state-of-the-art competitors according to several retrieval metrics but also runs much faster than most of them. [ABSTRACT FROM AUTHOR]

Boolean functions, coding theory and $t$ -designs have close connections and interesting interplay. A standard approach to constructing $t$ -designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting $t$ -designs. However, some linear codes hold $t$ -designs, although they do not satisfy the conditions in the Assmus-Mattson Theorem and do not admit a $t$ -transitive or $t$ -homogeneous group as a subgroup of their automorphisms. The major objective of this paper is to develop a theory for explaining such codes and obtaining such new codes and hence new $t$ -designs. To this end, a general theory for punctured and shortened codes of linear codes supporting $t$ -designs is established, a generalized Assmus-Mattson theorem is developed, and a link between 2-designs and differentially $\delta $ -uniform functions and 2-designs is built. With these general results, binary codes with new parameters and explicit weight distributions are obtained, new 2-designs and Steiner system $S(2, 4, 2^{n})$ are produced in this paper. [ABSTRACT FROM AUTHOR]

The quad-phase codes of a given length may be organized into equivalence classes relative to a set of operations preserving autocorrelation peak sidelobe level (PSL). Knowledge about these equivalence classes can be exploited for faster searches or for efficient listing of low- $\rm PSL$ codes. This paper is part of an effort to enumerate autocorrelation $\rm PSL$ equivalence classes for quad-phase codes of any length. An earlier paper achieved a formula for the number of $\rm PSL$ equivalence classes for even lengths $N \geq 2$ , as well as a simple algorithm for generating equivalence class representatives. This paper completes the enumeration task by extending the same formula to the odd lengths $N \geq 3$. The results make use of work by Yurramendi in enumerating equivalence classes for binary rectangular grids relative to horizontal reflection and vertical reflection. In addition, a method is described for generating a minimal set of $\rm PSL$ equivalence class representatives covering the space of quad-phase codes of any odd length $N > 3$. [ABSTRACT FROM AUTHOR]

This paper investigates the construction of rank-metric codes with specified Ferrers diagram shapes. These codes play a role in the multilevel construction for subspace codes. A conjecture from 2009 provides an upper bound for the dimension of a rank-metric code with given specified Ferrers diagram shape and rank distance. While the conjecture in its generality is wide open, several cases have been established in the literature. This paper contributes further cases of Ferrers diagrams and ranks for which the conjecture holds true. In addition, the proportion of maximal Ferrers diagram codes within the space of all rank-metric codes with the same shape and dimension is investigated. Special attention is being paid to MRD codes. It is shown that for growing field size the limiting proportion depends highly on the Ferrers diagram. For instance, for $[m\times 2]$ -MRD codes with rank 2 this limiting proportion is close to $1/e$. [ABSTRACT FROM AUTHOR]

The problem of universal source coding for binary trees is considered. Zhang, Yang, and Kieffer derived upper bounds on the average-case redundancy of codes based on directed acyclic graph (DAG) compression for binary tree sources with certain properties. In this paper, a natural class of binary tree sources is presented such that the demanded properties are fulfilled. Moreover, for both subclasses considered in the paper of Zhang, Yang, and Kieffer, their result is improved by deriving bounds on the maximal pointwise redundancy (or worst-case redundancy) instead of the average-case redundancy. Finally, using context-free tree grammars instead of DAGs, upper bounds on the maximal pointwise redundancy for certain binary tree sources are derived. This yields universal codes for new classes of binary tree sources. [ABSTRACT FROM AUTHOR]

Due to their high retrieval efficiency and low storage cost, cross-modal hashing methods have attracted considerable attention. Generally, compared with shallow cross-modal hashing methods, deep cross-modal hashing methods can achieve a more satisfactory performance by integrating feature learning and hash codes optimizing into a same framework. However, most existing deep cross-modal hashing methods either cannot learn a unified hash code for the two correlated data-points of different modalities in a database instance or cannot guide the learning of unified hash codes by the feedback of hashing function learning procedure, to enhance the retrieval accuracy. To address the issues above, in this paper, we propose a novel end-to-end Deep Cross-Modal Hashing with Hashing Functions and Unified Hash Codes Jointly Learning (DCHUC). Specifically, by an iterative optimization algorithm, DCHUC jointly learns unified hash codes for image-text pairs in a database and a pair of hash functions for unseen query image-text pairs. With the iterative optimization algorithm, the learned unified hash codes can be used to guide the hashing function learning procedure; Meanwhile, the learned hashing functions can feedback to guide the unified hash codes optimizing procedure. Extensive experiments on three public datasets demonstrate that the proposed method outperforms the state-of-the-art cross-modal hashing methods. [ABSTRACT FROM AUTHOR]

Reed-Muller codes are error-correcting codes used in many areas related to coding theory, such as electrical engineering and computer science. The binary $r^{th}$ -order Reed-Muller code $RM(r,n)$ can be viewed as the set of all $n$ -variable Boolean functions of algebraic degree at most $r$. Despite the intense work on these codes, many problems are known to be hard (notably, determining their covering radius) and remain open to this day. Fourteen years ago, Carlet and Mesnager improved in [IEEE Transactions on Information Theory, “Improving the Upper Bounds on the Covering Radii of Binary Reed-Muller Codes”, 53(1), 2007] the upper bound on the covering radius of the Reed-Muller code of order 2, and they deduced improved upper bounds on the covering radii of the Reed-Muller codes of higher orders. Until 2021, these upper bounds remained the best ones in the literature. The Reed-Muller code $RM(n-3,n)$ , which corresponds to the dual of the Reed-Muller code $RM(2,n)$ , has attracted much attention. One of the main reasons is that it is precisely the code that has been considered to get the upper bounds derived by Carlet and Mesnager. Those upper bounds have been obtained thanks to the characterization of the codewords of the Reed-Muller code, whose Hamming weights are strictly less than 2.5 times the minimum distance $2^{n-r}$ due to Kasami, Tokura, and Azumi. Despite their impressive work in the seventieth, a more refined study and profound description of those codewords of ${RM}({n-3},{n})$ whose Hamming weight equals 16, and especially 18, seem necessary, as it could help us significantly in improving the covering radius of Reed-Muller codes. In this paper, we push further the known results on the Reed-Muller codes by focusing on the Reed-Muller code ${RM}({n-3},{n})$. We provide a classification of the codewords of weights 16 and 18 of the Reed-Muller code $RM(n-3,n)$. Our algebraic descriptions allow us to count the number of such codewords and to enumerate all of them explicitly. [ABSTRACT FROM AUTHOR]

Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weight anticodes. Each one of these families of diameter perfect codes raises some different questions. We consider some of these questions and leave lot of ground for further research. Finally, as a consequence, some $t$ -intersecting families related to the well-known Erdös-Ko-Rado theorem, are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we formulate the problem to cope with the transmission of extra bits over an existing coded transmission link (referred to as coded payload link) without any cost of extra transmission energy or extra bandwidth. This is possible since a gap to the channel capacity typically exists for a practical code. A new concept, termed as accessible capacity, is introduced to specify the maximum rate at which the superposition transmission of extra bits is reliable and has a negligible effect on the performance of the coded payload link. For a binary-input output-symmetric (BIOS) memoryless channel, the accessible capacity can be characterized as the difference between the channel capacity and the mutual information rate of the coded payload link, which can be numerically evaluated for very short payload codes. For a general payload code, we present a simple lower bound on the accessible capacity, given by the channel capacity minus the coding rate of the payload code. We then focus on the scenarios where low-density parity-check (LDPC) codes are implemented for the payload link. We propose to transmit extra bits by random superposition for encoding, and exhaustive search (with the aid of statistical learning) for decoding. We further propose, by establishing an auxiliary channel (called syndrome channel) induced from “zero-forcing” over the binary field, to transmit extra bits with structured codes such as repetition codes and first-order Reed-Muller (RM) codes. Numerical results show that up to 60 extra bits can be reliably transmitted along with a rate-1/2 LDPC code of length 8064. [ABSTRACT FROM AUTHOR]

Let $n (>3)$ be a prime number and ${\mathbb {F}}_{2^{n}}$ a finite field of $2^{n}$ elements. Let $L ={\mathbb {F}}_{2^{n}}\cup \{\infty \}$ be the support set and $g(x)$ an irreducible polynomial of degree 6 over ${\mathbb {F}}_{2^{n}}$. In this paper, we obtain an upper bound on the number of extended irreducible binary Goppa codes $\Gamma (L, g)$ of degree 6 and length $2^{n}+1$. [ABSTRACT FROM AUTHOR]

In this paper, we are tackling the weakly-supervised referring expression grounding task, for the localization of a referent object in an image according to a query sentence, where the mapping between image regions and queries are not available during the training stage. In traditional methods, an object region that best matches the referring expression is picked out, and then the query sentence is reconstructed from the selected region, where the reconstruction difference serves as the loss for back-propagation. The existing methods, however, conduct both the matching and the reconstruction approximately as they ignore the fact that the matching correctness is unknown. To overcome this limitation, a discriminative triad is designed here as the basis to the solution, through which a query can be converted into one or multiple discriminative triads in a very scalable way. Based on the discriminative triad, we further propose the triad-level matching and reconstruction modules which are lightweight yet effective for the weakly-supervised training, making it three times lighter and faster than the previous state-of-the-art methods. One important merit of our work is its superior performance despite the simple and neat design. Specifically, the proposed method achieves a new state-of-the-art accuracy when evaluated on RefCOCO (39.21 percent), RefCOCO+ (39.18 percent) and RefCOCOg (43.24 percent) datasets, that is 4.17, 4.08 and 7.8 percent higher than the previous one, respectively. The code is available at https://github.com/insomnia94/DTWREG. [ABSTRACT FROM AUTHOR]

Radar radio source (RRS) recognition plays an important role in the fields of military electronic support systems (ESM) and civilian autonomous driving. The rapid development of machine learning technology, especially deep learning, has effectively and efficiently improved RRS intelligent recognition performances when operating in the increasingly complex electromagnetic environment. However, the data sampling limitation and computation cost are still severe challenges in real RRS recognition scenarios. In this paper, we propose a novel network based on meta-transfer learning, called RRSARNet, to achieve effective adaptive RRS recognition in the context of low signal-to-noise ratio (SNR). First, by using the short-time Fourier transform, a six-type small samples RRS simulation dataset with different SNR levels is constructed. Then, a novel RRSARNet, based on metric learning, is proposed, which consists of a four-layer embedding module and a four-layer relational module. Finally, the RRS dataset is divided into training, supporting and testing subsets, which are used to train and test the RRSARNet in a meta-transfer learning method. Experiments on the RRS dataset show that the proposed RRSARNet can achieve an overall accuracy (OA) above 96% and 99% when the SNR is above −15 dB and −10 dB, respectively. Even when the SNR is -30 dB, OA can reach more than 70%. For 5-way 1-shot and 5-way 5-shot experiments, the inference time of an image is about 0.043 and 0.140 milliseconds, respectively. Besides, experiments on the RRS simulation dataset and the two benchmark datasets, the RRSARNet performs better or more competitive than many existing state-of-the-art technologies in terms of recognition accuracy. [ABSTRACT FROM AUTHOR]

Supervised cross-modal hashing has attracted much attention. However, there are still some challenges, e.g., how to effectively embed the label information into binary codes, how to avoid using a large similarity matrix and make a model scalable to large-scale datasets, how to efficiently solve the binary optimization problem. To address these challenges, in this paper, we present a novel supervised cross-modal hashing method, i.e., scalaBle Asymmetric discreTe Cross-modal Hashing, BATCH for short. It leverages collective matrix factorization to learn a common latent space for the labels and different modalities, and embeds the labels into binary codes by minimizing a distance-distance difference problem. Furthermore, it builds a connection between the common latent space and the hash codes by an asymmetric strategy. In the light of this, it can perform cross-modal retrieval and embed more similarity information into the binary codes. In addition, it introduces a quantization minimization term and orthogonal constraints into the optimization problem, and generates the binary codes discretely. Therefore, the quantization error and redundancy may be much reduced. Moreover, it is a two-step method, making the optimization simple and scalable to large-scale datasets. Extensive experimental results on three benchmark datasets demonstrate that BATCH outperforms some state-of-the-art cross-modal hashing methods in terms of accuracy and efficiency. [ABSTRACT FROM AUTHOR]

Deletion errors are notoriously difficult to correct. The capacity of the binary deletion channel is not yet fully known, and there is no completely satisfactory solution even to the specific problem of correcting only two deletion errors. In this paper, we show that polarization theory, along with the powerful methods of polar coding and successive-cancellation decoding, can be extended to channels with deletion errors. Our results further generalize to channels corrupted by insertion errors and/or combinations of insertions and deletions. It was recently proposed to correct deletion errors using polar codes designed for the BEC, precoded with a sufficiently powerful CRC code. In this approach, in order to correct ${d}$ deletions in a codeword of length ${n}$ , successive-cancellation decoding is run separately for each of the $\binom {n}{d}$ possible deletion patterns, while treating the deleted symbols as erasures. The resulting complexity of decoding is ${O}\bigl ({n}^{d+1}\log {n}\bigr)$. In contrast, we develop herein a natural generalization of successive-cancellation decoding to channels with deletion errors. Our algorithm is based on the recursive structure of polar codes, processing the channel output in ${m} = \log _{2} {n}$ decoding layers, just like the conventional successive-cancellation decoder. Every node in each decoding layer propagates its uncertainty about the deletion pattern to the nodes in the next layer and, eventually, the correct deletion pattern becomes visible at the decoder output, with high probability. This makes it possible to consider at most $\binom {d+2}{2}$ scenarios at each node, and the overall decoding complexity is only ${O}\bigl ({d}^{3} {n} \log {n}\bigr)$ , which scales polynomially rather than exponentially with the number of deletions. Based on the proposed algorithm, we also investigate the channel polarization phenomenon. We are able to prove weak polarization where the number of deletions ${d}$ is small, i.e. ${d}={o}({n})$. We also conjecture the strong polarization for when ${d}={O}(1)$ and provide some evidence to support this conjecture. The strong polarization, if proven, implies that our coding scheme achieves the capacity of any binary-input symmetric channel which furthermore introduces a constant number of deletions. [ABSTRACT FROM AUTHOR]