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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]

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]

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]

In a breakthrough, Hastings showed that there exist quantum channels whose classical Holevo capacity is superadditive i.e. more classical information can be transmitted by quantum encoding strategies entangled across multiple channel uses as compared to unentangled quantum encoding strategies. Hastings’ proof used Haar random unitaries to exhibit superadditivity. In this paper we show that a unitary chosen uniformly at random from an approximate $n^{2/3}$ -design gives rise to a quantum channel with superadditive classical Holevo capacity, where $n$ is the dimension of the unitary exhibiting the Stinespring dilation of the channel superoperator. We follow the geometric functional analytic approach of Aubrun, Szarek and Werner in order to prove our result. More precisely we prove a sharp Dvoretzky-like theorem stating that, with high probability under the choice of a unitary from an approximate $t$ -design, random subspaces of large dimension make a Lipschitz function take almost constant value. Such theorems were known earlier only for Haar random unitaries. We obtain our result by appealing to Low’s technique for proving concentration of measure for an approximate $t$ -design, combined with a stratified analysis of the variational behaviour of Lipschitz functions on the unit sphere in high dimension. The stratified analysis is the main technical advance of this work. Haar random unitaries require at least $\Omega (n^{2})$ random bits in order to describe them with good precision. In contrast, there exist exact $n^{2/3}$ -designs using only $O(n^{2/3} \log n)$ random bits. Thus, our work can be viewed as a partial derandomisation of Hastings’ result, and a step towards the quest of finding an explicit quantum channel with superadditive classical Holevo capacity. Finally we also show that for any $p > 1$ , approximate unitary $n^{1.7}$ -designs give rise to channels violating subadditivity of Rényi $p$ -entropy. In addition to stratified analysis, the proof of this result uses a new technique of approximating a monotonic differentiable function defined on a closed bounded interval and its derivative by moderate degree polynomials which should be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, a family of $[{q}^{2}-1, 4, {q}^{2}-{q}-2]$ cyclic codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2}-1,{q}^{2}-5,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of cyclic codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of $[{q}^{2}, 4, {q}^{2}-{q}-1]$ linear codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2},{q}^{2}-4,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of linear codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Five infinite families of 2-designs are also constructed with three families of linear codes of this paper. [ABSTRACT FROM AUTHOR]

In literatures, there are various constructions of frequency hopping sequence (FHS for short) sets with good Hamming correlations. Some papers employed only multiplicative groups of finite fields to construct FHS sets, while other papers implicitly used only additive groups of finite fields for construction of FHS sets. In this paper, we make use of both multiplicative and additive groups of finite fields simultaneously to present a construction of optimal FHS sets. The construction provides a new family of optimal $\left ({{q}^{{m}}-1,\frac {{q}^{{m}-{t}}-1}{{r}},{\it{ rq}}^{{t}};\frac {{q}^{{m}-{t}}-1}{{r}}+1}\right)$ frequency hopping sequence sets archiving the Peng-Fan bound. Thus, some FHS sets constructed in literatures using either multiplicative groups or additive groups of finite fields are included in our family. In addition, some other FHS sets can be obtained via the well-known recursive construction through one-coincidence sequence set. [ABSTRACT FROM AUTHOR]

This paper studies an $n$ -dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, when $n=1$ the capacity-achieving input distribution is discrete with finitely many mass points, and when $n>1$ the capacity-achieving input distribution is supported on finitely many concentric shells. However, due to the previous proof technique, not even a bound on the exact number of mass points/shells was available. This paper provides an alternative proof of the finiteness of the number mass points/shells of the capacity-achieving input distribution while producing the first firm bounds on the number of mass points and shells, paving an alternative way for approaching many such problems. The first main result of this paper is an order tight implicit bound which shows that the number of mass points in the capacity-achieving input distribution is within a factor of two from the number of zeros of the downward shifted capacity-achieving output probability density function. Next, this implicit bound is utilized to provide a first firm upper on the support size of optimal input distribution, an $O(\mathsf {A}^{2})$ upper bound where $\mathsf {A}$ denotes the constraint on the input amplitude. The second main result of this paper generalizes the first one to the case when $n>1$ , showing that, for each and every dimension $n\ge 1$ , the number of shells that the optimal input distribution contains is $O(\mathsf {A}^{2})$. Finally, the third main result of this paper reconsiders the case $n=1$ with an additional average power constraint, demonstrating a similar $O(\mathsf {A}^{2})$ bound. [ABSTRACT FROM AUTHOR]

A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making them fully accessible for everyone. Thus, the first part of the paper develops a direct presentation of the codes by means of parity-check matrices, and the decoding algorithm rests upon matrix and linear maps manipulations. The somewhat more sophisticated mathematical context (non-commutative rings) needed for the proof of the correctness of the decoding algorithm is postponed to the second part. A final section locates the Reed-Solomon skew-differential codes introduced here within the general context of codes defined by means of skew polynomial rings. [ABSTRACT FROM AUTHOR]

It is well known that the capacity region of an average transmit power constrained Gaussian Broadcast Channel (GBC) with independent noise realizations at the receivers is enlarged by the presence of causal noiseless feedback. When the noise variances at the receivers are identical, even passive feedback via independent memoryless Gaussian links can lead to a capacity region enlargement. The last fact remains true even when the feedback noise variance is very high, and available only from one of the receivers. While such capacity enlargements are feasible for several other feedback models in the Gaussian BC setting, it is also known that feedback does not change the capacity region for physically degraded broadcast channels. In this paper, we consider a two user GBC with independent noise realizations at the receivers, where the feedback links from the receivers are corrupted by independent additive Gaussian noise processes. We investigate the set of four noise variances, two forward and two feedback, for which no capacity enlargement is possible. A sharp characterization of this region is derived, i.e., any quadruple outside the presented region will lead to a capacity enlargement, whereas quadruples inside will leave the capacity region unchanged. Our results lead to the conclusion that when the forward noise variances are different, too noisy a feedback from one of the receivers alone is not always beneficial for enlarging the capacity region, be it from the stronger user or the weaker one, in sharp contrast to the case of equal forward noise variances. [ABSTRACT FROM AUTHOR]

Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code-division multiple-access (MC-CDMA) systems. They can support more users than perfect complementary sequence sets in MC-CDMA systems. It is desirable to design QCSSs with good parameters that are a trade-off of large set size, small periodic maximum magnitude correlation and small alphabet size. The main results are to construct new infinite families of QCSSs that all have small alphabet size and asymptotically optimal periodic maximum magnitude correlation. In this paper, we propose three new constructions of QCSSs using additive characters over finite fields. Notably, these QCSSs have new parameters and small alphabet sizes. Using the properties of characters and character sums, we determine their maximum periodic correlation magnitudes and prove that these QCSSs are asymptotically optimal with respect to the lower bound. [ABSTRACT FROM AUTHOR]

Let P be a partial order on [n] = 1,2, . . . , n, Fqn be the linear space of n-tuples over a finite field Fq and w be a weight on Fq. In this paper, we consider metrics on Fqn induced by chain orders P over [n] and weights w over Fq, and we determine the cardinality of all optimal anticodes and completely classify them. Moreover, we determine all diameter perfect codes for a set of relevant instances on the aforementioned metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of linear codes defined by characteristic functions of certain subsets of a finite field. We derive a sufficient and necessary condition for such a code to be a minimal linear code by a character-theoretical approach. We obtain new three-weight or four-weight minimal linear codes that do not satisfy the Ashikhmin-Barg condition by using partial difference sets. We show that our construction yields minimal linear codes that do not arise from cutting vectorial blocking sets, and also discuss their applications in secret sharing schemes. [ABSTRACT FROM AUTHOR]

BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers. 1) The dimension of narrow-sense BCH codes of length $((q^{m}-1)/{\lambda })$ with designed distance $2\leq \delta \leq (({q^{\lceil (m+1)/2 \rceil }-1})/(\lambda)+1)$ is settled, where $\lambda $ is any factor of $(q-1)$. 2) The weight distributions of two classes of narrow-sense BCH codes of length $(({q^{m}-1})/2)$ with designed distance $\delta =(({(q-1)q^{m-1}-q^{\lfloor (m-1)/2\rfloor }-1})/2)$ and $\delta =(({(q-1)q^{m-1}-q^{\lfloor (m+1)/2\rfloor }-1})/2)$ are determined. 3) The weight distribution of a class of BCH codes of length $((q^{m}-1)/({q-1}))$ is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the additivity phenomenon in the quantum dynamic capacity region of a quantum channel for trading the resources of classical communication, quantum communication, and entanglement. Understanding such an additivity property is important if we want to optimally use a quantum channel for general communication purposes. However, in a lot of cases, the channel one will be using only has an additive single or double resource capacity region, and it is largely unknown if this could lead to a strictly superadditive double or triple resource capacity region, respectively. For example, if a channel has additive classical and quantum capacities, can the classical-quantum capacity region be strictly superadditive? In this paper, we answer such questions affirmatively. We give proof-of-principle requirements for these channels to exist. In most cases, we can provide an explicit construction of these quantum channels. The existence of these superadditive phenomena is surprising in contrast to the result that the additivity of both classical-entanglement and classical-quantum capacity regions imply the additivity of the triple resource capacity region for a given channel. [ABSTRACT FROM AUTHOR]

Let X and Y be dependent random variables. This paper considers the problem of designing a scalar quantizer for Y to maximize the mutual information between the quantizer’s output and X, and develops fundamental properties and bounds for this form of quantization, which is connected to the log-loss distortion criterion. The main focus is the regime of low ${I}({X};{Y})$ , where it is shown that, if X is binary, a constant fraction of the mutual information can always be preserved using $\mathcal {O}(\log (1/{I}({X};{Y})))$ quantization levels, and there exist distributions for which this many quantization levels are necessary. Furthermore, for larger finite alphabets $2 < |\mathcal {X}| < \infty $ , it is established that an $\eta $ -fraction of the mutual information can be preserved using roughly $(\log (| \mathcal {X} | /{I}({X};{Y})))^{\eta \cdot (|\mathcal {X}| - 1)}$ quantization levels. [ABSTRACT FROM AUTHOR]

As a crucial technique for integrated circuits (IC) test response compaction, X-compact employs a special kind of codes called X-codes for reliable compressions of the test response in the presence of unknown logic values (Xs). From a combinatorial view point, Fujiwara and Colbourn introduced an equivalent definition of X-codes and studied X-codes of small weights that have good detectability and X-tolerance. An $({\it\text { m, n, d, x}})~\text {X}$ -code is an $\text {m}\times \text {n}$ binary matrix with column vectors as its codewords. The parameters ${\it\text { d, x}}$ correspond to the test quality of the code. In this paper, bounds and constructions for constant weighted X-codes are investigated. First, we obtain a general result on the maximum number of codewords n for an $({\it\text { m, n, d, x}})~\text {X}$ -code of weight w, and we further improve this lower bound for the case with $\text {x}=2$ and $\text {w}=3$ through the probabilistic method. Then, using tools from additive combinatorics and finite fields, we present some explicit constructions for constant weighted X-codes with $\text {d}=3,7$ and $\text {x}=2$ , which are optimal for the case when $\text {d}=3, \text {w}=4$ and nearly optimal for the case when $\text {d}=3,\text {w}=3$. We also consider a special class of X-codes introduced by Fujiwara and Colbourn and improve the best known lower bound on the maximum number of codewords for this kind of X-codes. [ABSTRACT FROM AUTHOR]

This paper derives consistency results for estimation in the finite direct sum of reproducing kernel Hilbert spaces (RKHS) for dependent data. The link between penalized and constrained estimation is established. We consider the relation between topological equivalent norms for direct sums of RKHS. These norms have different implications for estimation. Estimation in a ball of the RKHS defined by these norms essentially results in estimation with a ridge and Lasso penalty, respectively. A greedy algorithm for the solution of the estimation problem under these two norms is discussed for general loss functions. [ABSTRACT FROM AUTHOR]

We consider the problem of recovering a high-dimensional structured signal from independent Gaussian linear measurements each of which is quantized to $b$ bits. The focus is on a specific method of signal recovery that extends a procedure originally proposed by Plan and Vershynin for one-bit quantization to a multi-bit setting. At the heart of this paper is a characterization of the optimal trade-off between the number of measurements m$ and the bit depth per measurement b$ given a total budget of B = m \cdot b -error in estimating the signal. It turns out that the choice b = 1$ is optimal for estimating the unit vector (direction) corresponding to the signal for any level of additive Gaussian noise before quantization as well as for a specific model of adversarial noise, whereas in a noiseless setting the choice b = 2$ is optimal for estimating the direction and the norm (scale) of the signal. Moreover, Lloyd–Max quantization is shown to be an optimal quantization scheme with respect to -estimation error. Our analysis is corroborated by the numerical experiments showing nearly perfect agreement with our theoretical predictions. This paper is complemented by an empirical comparison to alternative methods of signal recovery. The results of that comparison point to a regime change depending on the noise level: in a low-noise setting, the approach under study falls short of more sophisticated competitors while being competitive in moderate- and high-noise settings. [ABSTRACT FROM PUBLISHER]

We study communication systems over the state-dependent channels in the presence of a malicious state-aware jamming adversary. The channel has a memoryless state with an underlying distribution. The adversary introduces a jamming signal into the channel. The message and the entire state sequence are known non-causally to both the encoder and the adversary. This state-aware adversary may choose an arbitrary jamming vector depending on the message and the state vector. Taking an arbitrarily varying channel (AVC) approach, we consider two setups, namely, the discrete memoryless Gel’fand–Pinsker AVC and the additive white Gaussian dirty paper (DP) AVC. We determine the randomized coding capacity of both the AVCs under a maximum probability of error criterion. Similar to other randomized coding setups, we show that the capacity is the same even under the average probability of error criterion. Though the adversary can choose an arbitrary vector jamming strategy, we prove that the adversary cannot affect the rate any worse than when it employs a memoryless strategy, which depends only on the instantaneous state. Thus, the AVC capacity characterization is given in terms of the capacity of the worst memoryless channels with state, induced by the adversary employing such memoryless jamming strategies. For the DP-AVC, it is further shown that among memoryless jamming strategies, none impact the communication more than a memoryless Gaussian jamming strategy which completely disregards the knowledge of the state. Thus, the capacity of the DP-AVC equals that of a standard additive white Gaussian noise (AWGN) channel with two independent sources of AWGN, i.e., the channel noise and the jamming noise. [ABSTRACT FROM PUBLISHER]

In practical applications, such as direct spread code division multiple access communications, space-time codes and compressed sensing, and codebooks with small inner-product correlation are required. It is extremely difficult to construct codebooks achieving the Levenshtein bound. In this paper, two new constructions of infinitely many codebooks with multiplicative characters of finite fields are presented. These constructions produce complex codebooks asymptotically achieving the Levenshtein bound and codebooks asymptotically achieving the Welch bound. The codebooks presented in this paper have new parameters. [ABSTRACT FROM PUBLISHER]

Let $P=([n],\leq _{P})$ be a poset on $[n] = \{1,2,\ldots,n\}$ , $\mathbb {F}_{q}^{n}$ be the linear space of $n$ -tuples over a finite field $\mathbb {F}_{q}$ and $w$ be a weight on $\mathbb {F}_{q}$. In this paper we consider metrics on $\mathbb {F}_{q}^{n}$ which are induced by posets over $[n]$ and weights over $\mathbb {F}_{q}$. Such family of metrics extend both the additive metric induced by the weight $w$ (when the poset is an anti-chain) and the poset metrics (when the weight is the Hamming weight). Furthermore, the pomset metrics is also a particular case of our construction, consequently, we provide a simpler approach to these metrics without using the multiset structure originally proposed. For the general case, we provide a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we (re)obtain the groups of linear isometries of the poset, pomset and additive metric spaces. When considering a chain order, we develop, for codes on these spaces, several of the invariants and properties found in the classical coding theory. Our construction of metrics based on partial orders and any weight over the base field, highlights the dependence of the poset metric over $\mathbb {F}_{q}^{n}$ with the Hamming metric on $\mathbb {F}_{q}$ and the additive property of its extension on $\mathbb {F}_{q}^{n}$. [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]

This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating $P\ast \mathcal {N}_\sigma $ , for $\mathcal {N}_\sigma \triangleq \mathcal {N}(0,\sigma ^{2} \mathrm {I}_{d})$ , by $\hat {P}_{n}\ast \mathcal {N}_\sigma $ under different statistical distances, where $\hat {P}_{n}$ is the empirical measure. We examine the convergence in terms of the Wasserstein distance, total variation (TV), Kullback-Leibler (KL) divergence, and $\chi ^{2}$ -divergence. We show that the approximation error under the TV distance and 1-Wasserstein distance ($\mathsf {W}_{1}$) converges at the rate $e^{O(d)} n^{-1/2}$ in remarkable contrast to a (typical) $n^{-\frac {1}{d}}$ rate for unsmoothed $\mathsf {W}_{1}$ (and $d\ge 3$). Similarly, for the KL divergence, squared 2-Wasserstein distance ($\mathsf {W}_{2}^{2}$), and $\chi ^{2}$ -divergence, the convergence rate is $e^{O(d)} n^{-1}$ , but only if $P$ achieves finite input-output $\chi ^{2}$ mutual information across the additive white Gaussian noise (AWGN) channel. If the latter condition is not met, the rate changes to $\omega \left ({n^{-1}}\right)$ for the KL divergence and $\mathsf {W}_{2}^{2}$ , while the $\chi ^{2}$ -divergence becomes infinite – a curious dichotomy. As an application we consider estimating the differential entropy $h(S+Z)$ , where $S\sim P$ and $Z\sim \mathcal {N}_\sigma $ are independent $d$ -dimensional random variables. The distribution $P$ is unknown and belongs to some nonparametric class, but n independently and identically distributed (i.i.d) samples from it are available. Despite the regularizing effect of noise, we first show that any good estimator (within an additive gap) for this problem must have a sample complexity that is exponential in d. We then leverage the above empirical approximation results to show that the absolute-error risk of the plug-in estimator converges as $e^{O(d)} n^{-1/2}$ , thus attaining the parametric rate in n. This establishes the plug-in estimator as minimax rate-optimal for the considered problem, with sharp dependence of the convergence rate both in n and d. We provide numerical results comparing the performance of the plug-in estimator to that of general-purpose (unstructured) differential entropy estimators (based on kernel density estimation (KDE) or k nearest neighbors (kNN) techniques) applied to samples of $S+Z$. These results reveal a significant empirical superiority of the plug-in to state-of-the-art KDE and kNN methods. As a motivating utilization of the plug-in approach, we estimate information flows in deep neural networks and discuss Tishby’s Information Bottleneck and the compression conjecture, among others. [ABSTRACT FROM AUTHOR]

The Z2s-additive codes are subgroups of Zn2s, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z4-linear Hadamard codes. However, when s > 2, the dimension of the kernel of Z2s-linear Hadamard codes of length 2t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is computed for s = 3. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once t ≥ 3 is fixed. In this case, the number of nonequivalent such codes is also established. [ABSTRACT FROM AUTHOR]

Lossless variable-length source coding with codeword cost is considered for general sources. The problem setting, where we impose on unequal costs on code symbols, is called the variable-length coding with codeword cost. In this problem, the infimum of average codeword cost have already been determined for general sources. On the other hand, the overflow probability, which is defined as the probability of codeword cost being above a threshold, have not been considered yet. In this paper, we first determine the infimum of achievable threshold in the first-order sense and the second-order sense for general sources with additive memoryless codeword cost. Then, we compute it for some special sources such as i.i.d. sources and mixed sources. A generalization of the codeword cost is also discussed. [ABSTRACT FROM AUTHOR]

This paper explores the network simplification problem in the context of Gaussian half-duplex diamond networks. Specifically, given an N-relay diamond network, this problem seeks to derive fundamental guarantees on the capacity of the best N-relay subnetwork, as a function of the full network capacity. Simplification guarantees are presented in terms of a particular approximate capacity, termed Independent-Gaussian (IG) approximate capacity, that characterizes the network capacity to within an additive gap, which is independent of the channel coefficients and operating SNR. The main focus of this work is when k = N-1 relays are selected out of N relays in a diamond network. First, a simple algorithm is proposed which selects all relays except the one with the minimum IG approximate half-duplex capacity. It is shown that the selected (N-1) -relay subnetwork has an IG approximate half-duplex capacity that is at least 1/2 of the IG approximate half-duplex capacity of the full network and that for the proposed algorithm, this guarantee is tight. Furthermore, this work proves the following tight fundamental guarantee: there always exists a subnetwork of k = N-1 relays that have an IG approximate half-duplex capacity that is at least equal to (N-1)/N of the IG approximate half-duplex capacity of the full network. Finally, these results are extended to derive lower bounds on the fraction guarantee when k ∈ [1:N] relays are selected. The key steps in the proofs lie in the derivation of properties of submodular functions, which provide a combinatorial handle on the network simplification problem for Gaussian half-duplex diamond networks. [ABSTRACT FROM AUTHOR]

A user wants to retrieve a file from a database without revealing the identity of the file retrieved to the operator of the database (server), which is known as the problem of private information retrieval (PIR). If it is further required that the user obtains no information about the other files in the database, the concept of symmetric PIR (SPIR) is introduced to guarantee privacy for both parties. For SPIR, the server(s) need to access some randomness independent of the database, to protect the content of undesired files from the user. The information-theoretic capacity of SPIR is defined as the maximum number of information bits of the desired file retrieved per downloaded bit. In this paper, the problem of SPIR is studied for a distributed storage system with $N$ servers (nodes), where all data (including the files and the randomness) are stored in a distributed way. Specifically, the files are stored by an $(N,K_{C})$ -MDS storage code. The randomness is distributedly stored such that any $K_{C}$ servers store independent randomness information. We consider two scenarios regarding to the ability of the storage nodes to cooperate. In the first scenario considered, the storage nodes do not communicate or collude. It is shown that the SPIR capacity for MDS-coded storage (hence called MDS-SPIR) is $1-\frac {K_{C}}{N}$ , when the amount of the total randomness of distributed nodes (unavailable at the user) is at least $\frac {K_{C}}{N - K_{\vphantom {R_{l}}C}}$ times the file size. Otherwise, the MDS-SPIR capacity equals zero. The second scenario considered is the $T$ -colluding SPIR problem (hence called TSPIR). Specifically, any $T$ out of $N$ servers may collude, that is, they may communicate their interactions with the user to guess the identity of the requested file. In the special case with $K_{C}=1$ , i.e., the database is replicated at each node, the capacity of TSPIR is shown to be $1-\frac {T}{N}$ , with the ratio of the total randomness size relative to the file size be at least $\frac {T}{\vphantom {R_{l}}N - T}$. For TSPIR with MDS-coded storage (called MDS-TSPIR for short), when restricted to schemes with additive randomness where the servers add the randomness to the answers regardless of the queries received, the capacity is proved to equal $1-\frac {K_{C} + T - 1}{N}$ , with total randomness at least $\frac {K_{C} + T - 1}{N - K_{\vphantom {R_{l}}C} - T + 1}$ times the file size. The MDS-TSPIR capacity for general schemes remains an open problem. [ABSTRACT FROM AUTHOR]

New capacity upper bounds are presented for the discrete-time Poisson channel with no dark current and an average-power constraint. These bounds are a consequence of techniques developed for the seemingly unrelated problem of upper bounding the capacity of binary deletion and repetition channels. Previously, the best known capacity upper bound in the regime where the average-power constraint does not approach zero was due to Martinez (JOSA B, 2007), which is re-derived as a special case of the framework developed in this paper. Furthermore, this framework is carefully instantiated in order to obtain a closed-form bound that improves the result of Martinez everywhere. Finally, capacity-achieving distributions for the discrete-time Poisson channel are studied under an average-power constraint and/or a peak-power constraint and arbitrary dark current. In particular, it is shown that the support of the capacity-achieving distribution under an average-power constraint must only be countably infinite. This settles a conjecture of Shamai (IEE Proceedings I, 1990) in the affirmative. Previously, it was only known that the support must be an unbounded set. [ABSTRACT FROM AUTHOR]

An $(\boldsymbol {M},\boldsymbol {K},\boldsymbol {N},\delta _{\max })$ -quasi-complementary sequence set (QCSS) is referred to as a set of $\boldsymbol {M}$ 2-D matrices of order $\boldsymbol {K}\times \boldsymbol {N}$ with periodic tolerance $\delta _{\max }$. In a multicarrier code-division multiple-access (MC-CDMA) communication system, the set size $\boldsymbol {M}$ of a QCSS is equal to the maximum number of users it can support, and the periodic tolerance $\delta _{\max }$ determines the interference performance. For the application of a QCSS, it is desirable that the set size should be as large as possible, and the periodic tolerance should be as small as possible. In this paper, a framework of a periodic QCSS is proposed from additive and multiplicative characters associated with a specific integer set. It is then discovered that the parameters of the obtained QCSS are determined by the employed integer set. From this discovery, new classes of periodic QCSSs with large set sizes and low periodic tolerances are constructed and associated with some known integer sets. [ABSTRACT FROM AUTHOR]

Given two discrete random variables $X$ and $Y$ , with probability distributions $p=(p_{1}, \ldots , p_{n})$ and $q=(q_{1}, \ldots , q_{m})$ , respectively, let us denote by ${\mathcal{ C}}(p, q)$ the set of all couplings of $p$ and $q$ , that is, the set of all bivariate probability distributions that have $p$ and $q$ as marginals. In this paper, we study the problem of finding a joint probability distribution in ${\mathcal{ C}}(p, q)$ of minimum entropy (equivalently, a coupling that maximizes the mutual information between $X$ and $Y$), and we discuss several situations where the need for this kind of optimization naturally arises. Since the optimization problem is known to be NP-hard, we give an efficient algorithm to find a joint probability distribution in ${\mathcal{ C}}(p, q)$ with entropy exceeding the minimum possible at most by 1 bit, thus providing an approximation algorithm with an additive gap of at most 1 bit. Leveraging on this algorithm, we extend our result to the problem of finding a minimum-entropy joint distribution of arbitrary $k\geq {2}$ discrete random variables $X_{1}, \ldots , X_{k}$ , consistent with the known $k$ marginal distributions of the individual random variables $X_{1}, \ldots , X_{k}$. In this case, our algorithm has an additive gap of at most $\log k$ from optimum. We also discuss several related applications of our findings and extensions of our results to entropies different from the Shannon entropy. [ABSTRACT FROM AUTHOR]

In this paper, two different Gray-like maps from $Z_{p}^\alpha \times Z_{p^{k}}^\beta $ , where $p$ is prime, to $Z_{p}^{n}$ , $n={\alpha +\beta p^{k-1}}$ , denoted by $\varphi $ and $\Phi $ , respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if $C$ is a $Z_{p} Z_{p^{k}}$ -additive code, and $C^\bot $ is its dual, then the weight enumerators of the image $p$ -ary codes $\varphi (C)$ and $\Phi (C^\bot)$ are formally dual. This is a partial generalization of [D. S. Krotov, On $Z_{2^{k}}$ -dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532–1537], and the result is generalized to odd characteristic $p$ and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed $Z_{p} Z_{p^{2}}\ldots Z_{p^{k}}$ alphabet is given. [ABSTRACT FROM AUTHOR]

BCH codes are an interesting type of cyclic codes and have wide applications in communication and storage systems. Generally, it is very hard to determine the minimum distances of BCH codes. In this paper, we determine the weight distributions of two classes of primitive BCH codes $\mathcal C_{(q, m, \delta _{2})}$ and $\mathcal C_{(q, m, \delta _{3})}$ and their extended codes, which solve two problems proposed by Ding et al. It is shown that the extended codes $\overline {\mathcal C}_{(q, m, \delta _{2})}$ have four nonzero weights. We also employ the Hartmann-Tzeng bound to present the minimum distance of the dual code $\mathcal C_{(q, m, \delta _{2})}^\perp $ for $q \ge 5$. Inspired by the idea, we then determine the dimensions of a class of cyclic codes and give lower bounds on their minimum distances, which is greatly improved comparing with the BCH bound. Some optimal codes are obtained. [ABSTRACT FROM AUTHOR]

A $d$ -multiplicative secret sharing ($d$ -MSS) scheme allows the players to multiply $d$ shared secrets without recovering the secrets by converting their shares locally into an additive sharing of the product. It has been proved that the $d$ -MSS among $n$ players is possible if and only if no $d$ unauthorized sets of players cover the whole set of players (type $Q_{d}$). Although this result implies some limitations on SS in the context of MPC, the $d$ -multiplicative property is still useful for simplifying complex tasks of MPC by computing the product of $d$ field elements directly and non-interactively without any setup. This paper aims to improve the usefulness of the $d$ -MSS by enhancing the security against malicious adversaries. First, we introduce the notion of verifiably multiplicative SS, verifiably MSS for short, which is mainly formalized for detecting malicious behaviors. Informally, an SS scheme is verifiably $d$ -multiplicative if the scheme is $d$ -multiplicative and further enables the players to locally generate a share of a proof that the summed value is correct (i.e., the product of $d$ shared secrets). Secondly, we prove that there is no error-free verifiably MSS scheme whose decoder of the proof is additive, and that by accepting an error probability that can be chosen arbitrarily, there exists a verifiably $d$ -MSS scheme realizing a given access structure if and only if the access structure is of type $Q_{d}$. In the proposed construction, each share of a proof consists of only two field elements. This result means that we can efficiently achieve the optimal resiliency of the standard $d$ -MSS even against malicious adversaries. We note that by allowing a general class of decoders that includes a linear one, there is an error-free verifiably $d$ -MSS scheme if the access structure is of type $Q_{d+1}$. Finally, we generalize the $d$ -multiplicative property to a $d$ -or-less version where the number $d'$ of multiplied secrets with $d'\leq d$ is not known in advance. We show that a $d$ -or-less MSS scheme can be constructed from any $d$ -MSS scheme of the same access structure with a constant overhead, and the feasibility of (verifiably) $d$ -MSS implies that of (verifiably) $d$ -or-less MSS. [ABSTRACT FROM AUTHOR]

Estimation of Shannon and Rényi entropies of unknown discrete distributions is a fundamental problem in statistical property testing. In this paper, we give the first quantum algorithms for estimating $\alpha $ -Rényi entropies (Shannon entropy being 1-Rényi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for $\alpha $ -Rényi entropy estimation for all $\alpha \geq 0$ values, including tight bounds for the Shannon entropy, the Hartley entropy ($\alpha =0$), and the collision entropy ($\alpha =2$). We also provide quantum upper bounds for estimating min-entropy ($\alpha =+\infty $) as well as the Kullback–Leibler divergence. We complement our results with quantum lower bounds on $\alpha $ -Rényi entropy estimation for all $\alpha \geq 0$ values. Our approach is inspired by the pioneering work of Bravyi, Harrow, and Hassidim (BHH); however, with many new technical ingredients: 1) we improve the error dependence of the BHH framework by a fine-tuned error analysis together with Montanaro’s approach to estimating the expected output of quantum subroutines for $\alpha =0,1$ ; 2) we develop a procedure, similar to cooling schedules in simulated annealing, for general $\alpha \geq 0$ , and 3) in the cases of integer $\alpha \geq 2$ and $\alpha =+\infty $ , we reduce the entropy estimation problem to the $\alpha $ -distinctness and $\lceil \log n\rceil $ -distinctness problems, respectively. [ABSTRACT FROM AUTHOR]

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. The objective of this paper is to present some sufficient and necessary conditions that linear codes and cyclic codes have one-dimensional hull. It is shown that there are no such binary or ternary cyclic codes. Based on these characterizations, some constructions of linear codes with one-dimensional hull were given by employing quadratic number fields, partial difference sets, and difference sets. We also construct cyclic codes with one-dimensional hull. Some optimal codes with one-dimensional hull are obtained. [ABSTRACT FROM AUTHOR]

Consider a multi-variate time series ${(X_{t})}_{t=0}^{T}$ , where $X_{t} \in \mathbb {R}^{d}$ which may represent spike train responses for multiple neurons in a brain, crime event data across multiple regions, and many others. An important challenge associated with these time series models is to estimate an influence network between the $d$ variables, especially when $d$ is large, meaning we are in the high-dimensional setting. Prior work has focused on parametric vector auto-regressive models. However, parametric approaches are somewhat restrictive in practice. In this paper, we use the non-parametric sparse additive model (SpAM) framework to address this challenge. Using a combination of $\beta $ and $\phi $ -mixing properties of Markov chains and empirical process techniques for reproducing kernel Hilbert spaces (RKHSs), we provide upper bounds on mean-squared error in terms of the sparsity $s$ , the logarithm of the dimension $\log ~d$ , the number of time points $T$ , and the smoothness of the RKHSs. Our rates are sharp up to logarithm factors in many cases. We also provide numerical experiments that support our theoretical results and display the potential advantages of using our non-parametric SpAM framework for a Chicago crime data set. [ABSTRACT FROM AUTHOR]

In this paper, we establish some criteria to decide when a discrete memoryless channel admits a metric in such a way that the maximum likelihood decoding coincides with the nearest neighbor decoding. In particular, we prove a conjecture presented by M. Firer and J. L. Walker, establishing that every binary asymmetric channel admits a matched metric. [ABSTRACT FROM AUTHOR]

In this paper, we present a new family of maximum rank-distance (MRD) codes in $\mathbb F_{q}^{2n\times 2n}$ of minimum distance $2\leq d\leq 2n$. In particular, when $d=2n$ , we can show that the corresponding semifield is exactly a Hughes–Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to $\mathbb F_{q^{n}}$. We also prove that the MRD codes of minimum distance $2< d< 2n$ in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined. [ABSTRACT FROM AUTHOR]

Codebooks with low-coherence have wide utilization in many fields, such as direct spread code division multiple access communications, compressed sensing and so on. There are two major ingredients in this paper. The first is to present a new character sum, the hyper Eisenstein sum and study the properties of this character sum. As an application, the second ingredient is to propose two constructions of codebooks with the hyper Eisenstein sum. The codebooks generated by these constructions asymptotically meet the Welch bound. The parameters of these codebooks are new. [ABSTRACT FROM AUTHOR]

This paper shows that for the two-user Gaussian interference channel (G-IC) treating interference as noise without time sharing (TINnoTS) achieves the closure of the capacity region to within either a constant gap, or to within a gap of the order O(\log (\ln (\min ( \mathsf S, \mathsf I))/\gamma )) up to a set of Lebesgue measure \gamma \in (0,1] , where {\mathsf {S}} is the largest signal to noise ratio on the direct links and {\mathsf {I}}$ is the largest interference to noise ratio on the cross links. As a consequence, TINnoTS is optimal from a generalized degrees of freedom (gDoF) perspective for all channel gains except for a subset of zero measure. TINnoTS with Gaussian inputs is known to be optimal within 1/2 bit for a subset of the weak interference regime. Rather surprisingly, this paper shows that TINnoTS is gDoF optimal in all parameter regimes, even in the strong and very strong interference regimes where joint decoding of Gaussian inputs is optimal. For approximate optimality of TINnoTS in all parameter regimes, it is critical to use non-Gaussian inputs. This paper thus proposes to use mixed inputs as channel inputs for the G-IC, where a mixed input is the sum of a discrete and a Gaussian random variable. Interestingly, with reference to the Han–Kobayashi achievable scheme, the discrete part of a mixed input is shown to effectively behave as a common message in the sense that, although treated as noise, its effect on the achievable rate region is as if it were jointly decoded together with the desired messages at a non-intended receiver. The practical implication is that a discrete interfering input is a friend, while an Gaussian interfering input is in general a foe. This paper also discusses other practical implications of the proposed TINnoTS scheme with mixed inputs. Since TINnoTS requires neither explicit joint decoding nor time sharing, the results of this paper are applicable to a variety of oblivious or asynchronous channels, such as the block asynchronous G-IC (which is not an information stable channel) and the G-IC with partial codebook knowledge at one or more receivers. [ABSTRACT FROM AUTHOR]

A classical method of constructing a linear code over \mathrm GF(q) with a $t$ -design is to use the incidence matrix of the $t$ -design as a generator matrix over \mathrm GF(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 2-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterization of highly nonlinear Boolean functions is presented. [ABSTRACT FROM AUTHOR]

Motivated by applications in molecular biology and genotyping, this paper proposes a novel model of group testing for identifying items with real-valued status using nonbinary pooling designs under standard arithmetic observation. The purpose is to learn more information of each item to be tested rather than identify only which ones are defectives as was done in conventional group testing. This paper provides several efficiently decodable nonadaptive strategies for the considered problem. The major tool is a new structure called $q$ -ary additive $(w,d)$ -disjunct matrix, which is related to known structures: 1) the conventional disjunct matrix by Kautz and Singleton 1964 and 2) the SQ-disjunct matrix by Emad and Milenkovic 2012. [ABSTRACT FROM AUTHOR]

In a series of remarkable papers, Babadi and Tarokh proved the randomness of matrices and product of two matrices arising from binary linear block codes with respect to the empirical spectral distribution, provided that their dual distances are sufficiently large. However, numerical experiments conducted by Babadi and Tarokh revealed that Gold codes, which have a dual distance of 5, also possess such a randomness property. Hence, the interesting question was raised as to whether or not the stringent requirement of large dual distances can be relaxed in the theorems in order to explain the mysterious randomness of Gold sequences. In this paper, we improve the results of Babadi and Tarokh on several fronts and provide an affirmative answer to this question. [ABSTRACT FROM PUBLISHER]

A coding theorem and converse are proved for a large-class of abstract stationary channels with time structure including the result by Kadota and Wyner (1972) on continuous-time real-valued channels as special cases. As main contribution, the coding theorem is proved for a significantly weaker condition on the channel output memory – called total ergodicity with respect to finite alphabet block-memoryless input sources – and under a crucial relaxation of the measurability requirement for the channel. These improvements are achieved by introducing a suitable characterization of information rate capacity. It is shown that the $\psi$ -mixing output memory condition used by Kadota and Wyner is quite restrictive and excludes important channel models, in particular for the class of Gaussian channels. In fact, it is proved that for Gaussian (e.g., fading or additive noise) channels, the $\psi$ -mixing condition is equivalent to finite output memory. Furthermore, it is demonstrated that the measurability requirement of Kadota and Wyner is not satisfied for relevant continuous-time channel models such as linear filters, whereas the condition used in this paper is satisfied for these models. Moreover, a weak converse is derived for all stationary channels with time structure. Intersymbol interference as well as input constraints are taken into account in a general and flexible way, including amplitude and average power constraints as special case. Formulated in rigorous mathematical terms complete, explicit, and transparent proofs are presented. As a side product a gap in the proof of Kadota and Wyner – illustrated by a counterexample – is closed by providing a corrected proof of a lemma on the monotonicity of some sequence of normalized mutual information quantities. An operational perspective is taken, and an abstract framework is established, which allows to treat discrete- and continuous-time channels with (possibly infinite input and output) memory and arbitrary alphabets simultaneously in a unified way. [ABSTRACT FROM AUTHOR]

This paper addresses the correspondence between linear inequalities for Shannon entropy and differential entropy for sums of independent group-valued random variables. We show that any balanced (with the sum of coefficients being zero) linear inequality for Shannon entropy holds if and only if its differential entropy counterpart also holds; moreover, any linear inequality for differential entropy must be balanced. In particular, our result shows that recently proved differential entropy inequalities by Kontoyiannis and Madiman can be deduced from their discrete counterparts due to Tao in a unified manner. Generalizations to certain abelian groups are also obtained. Our proof of extending inequalities for Shannon entropy to differential entropy relies on a result of Rényi which relates the Shannon entropy of a finely discretized random variable to its differential entropy and also helps in establishing that the entropy of the sum of quantized random variables is asymptotically equal to that of the quantized sum; the converse uses the asymptotics of the differential entropy of convolutions with weak additive noise. [ABSTRACT FROM PUBLISHER]

This paper examines fundamental error characteristics for a general class of matrix completion problems, where the matrix of interest is a product of two a priori unknown matrices, one of which is sparse, and the observations are noisy. Our main contributions come in the form of minimax lower bounds for the expected per-element squared error for this problem under several common noise models. Specifically, we analyze scenarios where the corruptions are characterized by additive Gaussian noise or additive heavier-tailed (Laplace) noise, Poisson-distributed observations, and highly-quantized (e.g., one bit) observations, as instances of our general result. Our results establish that the error bounds derived in (Soni et al., 2016) for complexity-regularized maximum likelihood estimators achieve, up to multiplicative constants and logarithmic factors, the minimax error rates in each of these noise scenarios, provided that the nominal number of observations is large enough, and the sparse factor has (on an average) at least one non-zero per column. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce new Stein identities for gamma target distribution as well as a new non-linear channel specifically designed for gamma inputs. From these two ingredients, we derive an explicit and simple formula for the derivative of the input-output mutual information of this non-linear channel with respect to the channel quality parameter. This relation is reminiscent of the well-known link between the derivative of the input-output mutual information of additive Gaussian noise channel with respect to the signal-to-noise ratio and the minimum mean-square error. The proof relies on a rescaled version of De Bruijn identity for gamma target distribution together with a stochastic representation for the gamma-specific Fisher information. Finally, we are able to derive precise bounds and asymptotics for the input-output mutual information of the non-linear channel with gamma inputs. [ABSTRACT FROM AUTHOR]

In recent literature, a family of optimal linear locally recoverable codes (LRC codes) that attain the maximum possible distance (given code length, cardinality, and locality) is presented. The key ingredient for constructing such optimal linear LRC codes is the so-called $r$ -good polynomials, where r$ is equal to the locality of the LRC code. However, given a prime p$ , known constructions of r exist only for some special integers r$ , and the problem of constructing optimal LRC codes over small field for any given locality is still open. In this paper, by using function composition, we present two general methods of designing good polynomials, which lead to three new constructions of r$ -good polynomials. Such polynomials bring new constructions of optimal LRC codes. In particular, our constructed polynomials as well as the power functions yield optimal (n,k,r) for all positive integers $r$ as localities, where $q$ is near the code length $n$ . [ABSTRACT FROM AUTHOR]

Caching plays an important role in reducing the backbone traffic when serving high-volume multimedia content. Recently, a new class of coded caching schemes have received significant interest, because they can exploit coded multi-cast opportunities to further reduce backbone traffic. Without considering file popularity, prior works have characterized the fundamental performance limits of coded caching through a deterministic worst-case analysis. However, when heterogeneous file popularity is considered, there remain open questions regarding the fundamental limits of coded caching performance. In this paper, for an arbitrary popularity distribution, we first derive a new information-theoretic lower bound on the expected transmission rate of any coded caching schemes. We then show that a simple coded-caching scheme attains an expected transmission rate that is at most a constant factor away from the lower bound. Unlike other existing studies, the constant factor that we derived is independent of the popularity distribution. [ABSTRACT FROM PUBLISHER]

Linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator. After deriving the MacWilliams identities for these enumerators, we are able to prove algebraic linear programming bounds, such as the Singleton bound, the Hamming bound, and the first linear programming bound. Our Singleton bound and Hamming bound are more general than the previous bounds for entanglement-assisted quantum stabilizer codes. In addition, we show that the first linear programming bound improves the Hamming bound when the relative distance is sufficiently large. On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length. In particular, we show that there is no $[[27, 15, 5]]$ or $[[28, 14, 6]]$ stabilizer code. We also discuss the existence of some entanglement-assisted quantum stabilizer codes with maximal entanglement. As a result, the upper and lower bounds on the minimum distance of maximal-entanglement quantum stabilizer codes with length up to 20 are significantly improved. [ABSTRACT FROM PUBLISHER]