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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]

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]

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 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]

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]

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]

Bazzi and Mitter showed that binary dihedral group codes are asymptotically good. In this paper we prove that the dihedral group codes over any finite field with strong duality property are asymptotically good. If the characteristic of the field is even, self-dual dihedral group codes are asymptotically good. If the characteristic of the field is odd, maximal self-orthogonal dihedral group codes and LCD dihedral group codes are asymptotically good. [ABSTRACT FROM AUTHOR]

Consider a channel with a given input alphabet size and input distribution. Our aim is to degrade or upgrade it to a channel with at most L output letters. Channel Q is degraded with respect to a channel W if Q can be obtained from W by processing the output of W. Upgrading is the inverse relation. This paper contains four main results. The first result, from which the paper title is derived, deals with the so called “greedy–merge” algorithm. We derive an upper bound on the reduction in mutual information between input and output, as a function of L. This upper bound is within a constant factor of an algorithm-independent lower bound. Thus, we establish that greedy–merge is optimal in the power-law sense (i.e. the power of L). The other main results deal with upgrading. The second result shows that a certain sequence of channels, which was previously shown to be “hard” for degrading, displays the same hardness in the context of upgrading. That is, suppose we are given such a channel and a corresponding input distribution. If we upgrade (degrade) to a new channel with L output letters, we incur an increase (decrease) in mutual information between input and output. We show that a previously derived bound on the decrease in mutual information for the degrading case is also a lower bound on the increase for the upgrading case. The third result is an efficient algorithm for optimal upgrading, in the binary-input case. That is, we are given a channel and an input distribution. We must find an upgraded channel with L output letters, for which the increase in mutual information is minimal. We give a simple characterization of such a channel, which implies an efficient algorithm. The fourth result is an analog of the first result for the upgrading case when the input is binary. That is, we first present a sub-optimal algorithm for the setting considered in the third result. The main advantage of the sub-optimal algorithm is that it is amenable to analysis. We carry out the analysis and show that the increase incurred in mutual information is within a constant factor of the lower bound derived in the second result. [ABSTRACT FROM AUTHOR]

A primitive $k$ -batch code encodes a string $\boldsymbol {x}$ of length $n$ into a string $\boldsymbol {y}$ of length $N$ , such that each multiset of $k$ symbols from $\boldsymbol {x}$ has $k$ mutually disjoint recovering sets from $\boldsymbol {y}$. In this paper, we discuss new constructions of binary primitive batch codes. First, we develop novel explicit and random coding constructions of linear primitive batch codes based on finite geometries. Second, a new explicit coding construction of binary primitive batch codes based on bivariate lifted multiplicity codes is provided. For any $k=n^ \varepsilon $ with $\varepsilon \in (0,0.47)\setminus \{1/5,1/4\}$ , our proposed codes have a better trade-off between the redundancy and the parameters $k,n$ than previously known batch codes. [ABSTRACT FROM AUTHOR]

Various kinds of features prove to be effective for content-based image retrieval. However, due to the diversity of image contents, a descriptor may achieve impressive performance on specific images while becoming invalid on others. Although some efforts have been made to combine features as complementary counterparts, proper weighting scheme is still a challenge for fast and accurate retrieval. In this paper, we propose an effective fusion method, termed as Topo-correlation (Topo), where the importance of each feature is measured by cross-view correlations on local affinity graphs. Specifically, the weights of similarities are node-sensitive as well as modality-sensitive, thus boosting the results of good cues while depressing adverse factors for individual images. By estimating the consensus of similarity scores with regard to a query-driven criterion, the weighted graphs are generated efficiently with low computational complexity. Extensive experimental results on four benchmarks demonstrate the superiority of the proposed approach over the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

The binary Hamming codes with parameters $[{2^{m}-1, 2^{m}-1-m, 3}]$ are perfect. Their extended codes have parameters $[{2^{m}, 2^{m}-1-m, 4}]$ and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes with parameters $[{2^{m+s}+2^{s}-2^{m},2^{m+s}+2^{s}-2^{m}-2m-2,4}]$ , which have better information rates than the class of extended binary Hamming codes, and are also distance-optimal. The second objective is to construct a class of distance-optimal binary codes with parameters $[{2^{m}+2, 2^{m}-2m, 6}]$. Both classes of binary linear codes have new parameters. [ABSTRACT FROM AUTHOR]

MDS codes and self-dual codes are important families of classical codes in coding theory. Therefore, it is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field ${\mathbb F}_{q}$ is competely solved for $q$ is even. In the literature, there are many known constructions of MDS self-dual codes for $q$ is odd. In this paper, we present a unified approach on the existence of MDS self-dual codes with concise statements and simplified proof. It is illustrated that some of known results can also be stated in this framework. Furthermore, we can obtain some new MDS self-dual codes, especially for the case when $q$ is not a square. [ABSTRACT FROM AUTHOR]

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length $ {n} > {q}+1$ are also obtained. [ABSTRACT FROM AUTHOR]

The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions on the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As a consequence, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, an entanglement-assisted quantum error-correcting $[[21{s}+5,3,16{s}+3;21{s}+2]]$ code with maximal entanglement is constructed for the first time from a quaternary Hermitian linear complementary dual [26, 3, 19] code. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fundamentals of puncturing and shortening for polar codes, based on binary domination. For punctured and shortened polar codes, we prove that the partial order by binary domination completely determines both incapable bit patterns (by puncturing) and fixed bit patterns (by shortening). In particular, we give a necessary and sufficient condition for an encoder output bit to be fixed by additionally shortening a single encoder input bit, as well as a necessary and sufficient condition for an encoder input bit to be made incapable by additionally puncturing a single encoder output bit. We also identify all the puncturing bit patterns yielding a given incapable bit pattern. These results provide a guidance to design a practical rate-matching scheme for polar codes. As an example, we present a rate-matching scheme based on our analytical observations. Numerical results show that it performs well over a wide range of code lengths and rates, compared with conventional rate-matching schemes. [ABSTRACT FROM AUTHOR]

The zero-error capacity of a channel is defined as the maximum rate at which it is possible to transmit information with zero probability of error. In this paper, we settle all previously unsolved cases for the zero-error capacity of binary channels with one memory. [ABSTRACT FROM AUTHOR]

The projective special linear group ${\mathrm {PSL}}_{2}(n)$ is 2-transitive for all primes $n$ and 3-homogeneous for $n \equiv 3 \pmod {4}$ on the set $\{0,1, \ldots, n-1, \infty \}$. It is known that the extended odd-like quadratic residue codes are invariant under ${\mathrm {PSL}}_{2}(n)$. Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes $n \equiv 1 \pmod {4}$ , an infinite family of 3-designs for primes $n \equiv 3 \pmod {4}$. To construct more $t$ -designs with $t \in \{2, 3\}$ , one would search for other extended cyclic codes over finite fields that are invariant under the action of ${\mathrm {PSL}}_{2}(n)$. The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under ${\mathrm {PSL}}_{2}(n)$. [ABSTRACT FROM AUTHOR]

A source coding problem in which a central source has to satisfy the demands of several receivers, with each receiver having some subset of the messages (side-information) held by the source is considered. The source has knowledge of only the cardinality of the side-information at each receiver. The encoding scheme used by the source to transmit at a higher throughput is referred to as an informed source code. A technique to obtain informed source codes by using $\ell $ -th Near Maximum Distance Separable (Near-MDS) Codes is presented. The advantage of using $\ell $ -th Near-MDS codes is the reduction in field size required. For certain informed source coding problems, the code obtained from $\ell $ -th Near-MDS codes is shown to be of the minimum length under certain field size restrictions. The same technique can be used for a given index coding problem to obtain index codes. The index codes obtained through this technique are optimal for, but not limited to, special cases of index coding problems discussed in the paper. Finding an optimal solution to a general index coding problem is NP hard and this technique helps in finding binary suboptimal solutions. Using the Gilbert-Varshamov bound, an upper bound on the lengths of optimal binary informed source codes is obtained. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a new multi-edge metric-constrained quasi-cyclic progressive edge-growth algorithm (MM-QC-PEGA), which is suitable for constructing both single- and multi-weighted (binary) QC low-density parity-check (LDPC) codes with arbitrary lengths, rates, circulant sizes, and variable node (VN)-degree distributions. The MM-QC-PEGA is able to detect all cyclic-edge-set-minimum-virtual cycles (CMVCs), as it accurately computes the metric value of each CMVC with an a posteriori test. In addition, we propose a new greatest-common-divisor (GCD)-approximation for time efficiently approximating the metric value of a CMVC without a posteriori tests, and propose a new GCD-approximated MM-QC-PEGA (G-MM-QC-PEGA) by using the GCD-approximation to replace the a posteriori test involved in the MM-QC-PEGA. As a result, the G-MM-QC-PEGA is faster than the MM-QC-PEGA, and the CMVCs undetectable to the G-MM-QC-PEGA under multi- and single-weighted QC-LDPC code graphs have minimum lengths of eight and ten, respectively. Moreover, we propose a new masking technique that is efficient in masking the parity-check matrix consisting of an array of arbitrary circulants of the same size. Compared with the several existing works, our proposed algorithms could perform better or comparably in terms of avoiding generating small cycles and error performances. Our proposed algorithms somewhat perfect the works of QC-LDPC code construction. [ABSTRACT FROM PUBLISHER]

Switch codes, first proposed by Wang et al., are codes that are designed to increase the parallelism of data writing and reading processes in network switches. A network switch is required to write $n$ incoming packets and read $k$ outgoing packets while using $m$ memory banks, each able to write and read one packet per time unit. Each set of $n$ packets written to the switch simultaneously is called a generation. The objective is to store the packets in the banks such that every request of $k$ packets, which can belong to previous generations, can be handled by reading at most one packet from every bank. In this paper, we study a new type of switch codes that can simultaneously deliver large packet request and good coding rate. These attractive features are achieved by relaxing the request model to a natural sub-class we call consecutive requests. For this new request model, we define a new type of codes called consecutive switch codes. These codes are studied in both the computational and combinatorial models, corresponding to whether the data can be encoded or not. For binary codes, we also study an intermediate model in which a coded packet is formed by the XOR operations of at most two input packets. We present several code constructions and prove the optimality of one family of these codes by providing the corresponding lower bound. Finally, we introduce a construction of conventional switch codes, which improves upon the best known results for the case $n=k$ . [ABSTRACT FROM AUTHOR]

The Johnson-type upper bound on the maximum size of a code of length n , distance d=2w-1 , and constant composition \overline w is \lfloor \dfrac \vphantom R^.nw1\rfloor , where w is the total weight and w_{1} is the largest component of {\overline {w}} . Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let Nccc({\overline {w}}) be the smallest such length. The determination of Nccc({\overline {w}}) is trivial for binary codes. This paper provides a lower bound on Nccc({\overline {w}}) , which is shown to be tight for all ternary and quaternary codes by giving new combinatorial constructions. Consequently, by the refining method, we determine the values of Nccc({\overline {w}}) , for all $q$ -ary constant-composition codes, provided that 3w_{1}\geq w with finite possible exceptions. [ABSTRACT FROM AUTHOR]

Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over \mathop \mathrm GR\nolimits (2^a,m) of length 2^s have been characterized: the ring \mathcal R2(a,m,-1)= {\mathrm{ GR}}(2^{a},m)[x]/\langle x^{2^{s}}+1\rangle is a chain ring. Furthermore, these results have been generalized to \lambda -constacyclic codes for any unit \lambda of the form 4z-1 , . In this paper, we study more general cases and investigate all cases, where {\mathcal{ R}}_{p}(a,m,\gamma) = {\mathrm{ GR}}(p^{a},m)[x]/\langle x^{p^{s}}-\gamma \rangle is a chain ring. In particular, the necessary and sufficient conditions for the ring {\mathcal{ R}}_{p}(a,m,\gamma) to be a chain ring are obtained. In addition, by using this structure we investigate all \gamma -constacyclic codes over \mathop {\mathrm {GR}}\nolimits (p^{a},m) when {\mathcal{ R}}_{p}(a,m,\gamma)$ is a chain ring. The necessary and sufficient conditions for the existence of self-orthogonal and self-dual \gamma$ -constacyclic codes are also provided. Among others, for any prime p$ , the structure of is used to establish the Hamming and homogeneous distances of $\gamma$ -constacyclic codes. [ABSTRACT FROM PUBLISHER]

Multiply constant-weight codes (MCWCs) have been recently studied to improve the reliability of certain physically unclonable function response. In this paper, we give combinatorial constructions for the MCWCs, which yield several new infinite families of optimal MCWCs. Furthermore, we demonstrate that the Johnson-type upper bounds of the MCWCs are asymptotically tight for fixed Hamming weights and distances. Finally, we provide bounds and constructions of the 2-D MCWCs. [ABSTRACT FROM AUTHOR]

In this paper, we solve the open problems raised in [8] and present some examples to illustrate the obtained results. Moreover, we work out the diophantine problem by Shi and Wang and then give the sufficient conditions for the nonexistence of two-Lee weight projective codes over \mathbb Z4 with type 4^k12^{k2} . [ABSTRACT FROM PUBLISHER]

In this paper, let n=2k and d=3\cdot 2^{k}-2 with k\geq 3 and \gcd (d,2^n-1)=1 . Based on some analysis of certain equations over finite fields and the number of codewords with Hamming weight five in Zetterberg code, the correlation distribution between a binary m -sequence of period 2^n-1 and its $d$ -decimation sequence is completely determined. This solves a ten-year-old open problem proposed by Dobbertin et al. [ABSTRACT FROM PUBLISHER]

Projective Reed–Muller codes were introduced by Lachaud in 1988 and their dimension and minimum distance were determined by Serre and Sørensen in 1991. In coding theory, one is also interested in the higher Hamming weights, to study the code performance. Yet, not many values of the higher Hamming weights are known for these codes, not even the second lowest weight (also known as the next-to-minimal weight) is completely determined. In this paper, we determine all the values of the next-to-minimal weight for the binary projective Reed–Muller codes, which we show to be equal to the next-to-minimal weight of Reed–Muller codes in most, but not all, cases. [ABSTRACT FROM PUBLISHER]

Product quantization-based approaches are effective to encode high-dimensional data points for approximate nearest neighbor search. The space is decomposed into a Cartesian product of low-dimensional subspaces, each of which generates a sub codebook. Data points are encoded as compact binary codes using these sub codebooks, and the distance between two data points can be approximated efficiently from their codes by the precomputed lookup tables. Traditionally, to encode a subvector of a data point in a subspace, only one sub codeword in the corresponding sub codebook is selected, which may impose strict restrictions on the search accuracy. In this paper, we propose a novel approach, named optimized cartesian K-means (ock-means), to better encode the data points for more accurate approximate nearest neighbor search. In ock-means, multiple sub codewords are used to encode the subvector of a data point in a subspace. Each sub codeword stems from different sub codebooks in each subspace, which are optimally generated with regards to the minimization of the distortion errors. The high-dimensional data point is then encoded as the concatenation of the indices of multiple sub codewords from all the subspaces. This can provide more flexibility and lower distortion errors than traditional methods. Experimental results on the standard real-life data sets demonstrate the superiority over state-of-the-art approaches for approximate nearest neighbor search. [ABSTRACT FROM PUBLISHER]