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The size of the Hamming distance spectrum of a code has received great attention in recent research. The main objective of this paper is to extend these significant theories to the $b$-symbol distance spectrum. We examine this question for various types of codes, including unrestricted codes, additive codes, linear codes, and cyclic codes, successively. For the first three cases, we determine the maximum size of the $b$-symbol distance spectra of these codes smoothly. For the case of cyclic codes, we introduce three approaches to characterize the upper bound for the cardinality of the $b$-symbol weight spectrum of cyclic codes, namely the period distribution approach, the primitive idempotent approach, and the $b$-symbol weight formula approach. As two by-products of this paper, the maximum number of symplectic weights of linear codes is determined, and a basic inequality among the parameters $[n,k,d_H(\C)]_q$ of cyclic codes is provided.

The escalating size of Deep Neural Networks (DNNs) has spurred a growing research interest in hosting and serving DNN models across multiple devices. A number of studies have been reported to partition a DNN model across devices, providing device placement solutions. The methods appeared in the literature, however, either suffer from poor placement performance due to the exponential search space or miss an optimal placement as a consequence of the reduced search space with limited heuristics. Moreover, these methods have ignored the runtime inter-operator optimization of a computation graph when coarsening the graph, which degrades the end-to-end inference performance. This paper presents Moirai that better exploits runtime inter-operator fusion in a model to render a coarsened computation graph, reducing the search space while maintaining the inter-operator optimization provided by inference backends. Moirai also generalizes the device placement algorithm from multiple perspectives by considering inference constraints and device heterogeneity.Extensive experimental evaluation with 11 large DNNs demonstrates that Moirai outperforms the state-of-the-art counterparts, i.e., Placeto, m-SCT, and GETF, up to 4.28$\times$ in reduction of the end-to-end inference latency. Moirai code is anonymously released at \url{https://github.com/moirai-placement/moirai}.

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles [Discrete Math. 321, 2014] and [SIAM J. Discrete Math. 27(4), 2013], Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall provide answers to some open problems presented in the literature. The first one concerns Open Problem 1, proposed by Ding and Zhou in Discrete Math. 321, 2014. The two others are Open Problems 5.16 and 5.25, raised by Ding in [SIAM J. Discrete Math. 27(4), 2013].

The $b$-symbol read channel is motivated by the limitations of the reading process in high density data storage systems. The corresponding new metric is a generalization of the Hamming metric known as the $b$-symbol weight metric and has become an important object in coding theory. In this paper, the general $b$-symbol weight enumerator formula for irreducible cyclic codes is presented by using the Gaussian period and a new invariant $\#U(b,j,N_1)$. The related $b$-symbol weight hierarchies $\{d_1(\C),d_2(\C),\ldots,d_K(\C)\}$ ($K=\dim(\C)$) are given for some cases. The shortened codes which are optimal from some classes of irreducible cyclic codes are given, where the shorten set $\mathcal{T}$ is the complementary set of $b$-symbol support of some codeword with the minimal $b$-symbol weight.

It is a challenging task to predict financial markets. The complexity of this task is mainly due to the interaction between financial markets and market participants, who are not able to keep rational all the time, and often affected by emotions such as fear and ecstasy. Based on the state-of-the-art approach particularly for financial market predictions, a hybrid convolutional LSTM Based variational sequence-to-sequence model with attention (CLVSA), we propose a novel deep learning approach, named dual-CLVSA, to predict financial market movement with both trading data and the corresponding social sentiment measurements, each through a separate sequence-to-sequence channel. We evaluate the performance of our approach with backtesting on historical trading data of SPDR SP 500 Trust ETF over eight years. The experiment results show that dual-CLVSA can effectively fuse the two types of data, and verify that sentiment measurements are not only informative for financial market predictions, but they also contain extra profitable features to boost the performance of our predicting system., Comment: 8 pages, 2021 20th IEEE International Conference on Machine Learning and Applications (ICMLA). IEEE, 2021

The symbol-pair read channel was first proposed by Cassuto and Blaum. Later, Yaakobi et al. generalized it to the $b$-symbol read channel. It is motivated by the limitations of the reading process in high density data storage systems. One main task in $b$-symbol coding theory is to determine the $b$-symbol weight hierarchy of codes. In this paper, we study the $b$-symbol weight hierarchy of the Kasami codes, which are well known for their applications to construct sequences with optimal correlation magnitudes. The complete symbol-pair weight distribution of the Kasami codes is determined.

Recently, $b$-symbol codes are proposed to protect against $b$-symbol errors in $b$-symbol read channels. It is an interesting subject of study to consider the complete $b$-symbol weight distribution of cyclic codes since $b$-symbol metric is a generalization for Hamming metric. The complete $b$-symbol Hamming weight distribution of irreducible codes is known in only a few cases. In this paper, we give a complete $b$-symbol Hamming weight distribution of a class of irreducible codes with two nonzero $b$-symbol Hamming weights.

It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper., Comment: This manuscript was first submitted to IEEE Tran. Inf. Theory in 06, April, 2021

The $r$-th generalized Hamming metric and the $b$-symbol metric are two different generalizations of Hamming metric. The former is used on the wire-tap channel of Type II, and the latter is motivated by the limitations of the reading process in high-density data storage systems and applied to a read channel that outputs overlapping symbols. In this paper, we study the connections among the three metrics (that is, Hamming metric, $b$-symbol metric, and $r$-th generalized Hamming metric) mentioned above and give a conjecture about the $b$-symbol Griesmer Bound for cyclic codes. %Furthermore, we explore the combinatorial function of the size of the $b$-symbol weight set of a code $C$., Comment: Submitted on 6 July, 2021