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A key component of many graph neural networks (GNNs) is the pooling operation, which seeks to reduce the size of a graph while preserving important structural information. However, most existing graph pooling strategies rely on an assignment matrix obtained by employing a GNN layer, which is characterized by trainable parameters, often leading to significant computational complexity and a lack of interpretability in the pooling process. In this paper, we propose an unsupervised graph encoder–decoder model to detect abnormal nodes from graphs by learning an anomaly scoring function to rank nodes based on their degree of abnormality. In the encoding stage, we design a novel pooling mechanism, named LCPool, which leverages locality-constrained linear coding for feature encoding to find a cluster assignment matrix by solving a least-square optimization problem with a locality regularization term. By enforcing locality constraints during the coding process, LCPool is designed to be free from learnable parameters, capable of efficiently handling large graphs, and can effectively generate a coarser graph representation while retaining the most significant structural characteristics of the graph. In the decoding stage, we propose an unpooling operation, called LCUnpool, to reconstruct both the structure and nodal features of the original graph. We conduct empirical evaluations of our method on six benchmark datasets using several evaluation metrics, and the results demonstrate its superiority over state-of-the-art anomaly detection approaches. [ABSTRACT FROM AUTHOR]

In this paper we review existing hard-decision decoding algorithms for product codes along with different post-processing techniques used in conjunction with the iterative decoder for product codes. We improve the decoder by Reddy and Robinson and use it to create a new post-processing technique. The performance of this new post-processing technique is evaluated through simulations, and these suggest that our new post-processing technique outperforms previously known post-processing techniques which are not tailored for specific codes. The cost of using the new post-processing technique is that the algorithm becomes more complex. However, the post-processing is applied very rarely unless the channel is very noisy, and hence the increase in computational complexity is negligible for most choices of parameters. Finally, we propose a new algorithm that combines existing techniques in a way that avoids the error floor with short relatively high rate codes. The algorithm should also avoid the error floor with long high rate codes, but further work is needed to confirm this. [ABSTRACT FROM AUTHOR]

Linear codes with complementary duals, shortly named LCD codes, are linear codes whose intersection with their duals is trivial. In this paper, we outline a construction for LCD codes over finite fields from the adjacency matrices of two-class association schemes. These schemes consist of either strongly regular graphs (SRGs) or doubly regular tournaments (DRTs). Under certain conditions, the method yields formally self-dual codes. Further, we propose a decoding algorithm that can be feasible for the LCD codes obtained using one of the given methods. [ABSTRACT FROM AUTHOR]

As a subclass of linear codes, cyclic codes have efficient encoding and decoding algorithms, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, we give a general construction of optimal p-ary cyclic codes which leads to three explicit constructions. In addition, another class of p-ary optimal cyclic codes are presented. [ABSTRACT FROM AUTHOR]

Due to efficient encoding and decoding algorithms, BCH codes have many applications in data storage systems and satellite communications. Meanwhile, one advantage of BCH codes is to construct quantum codes. In this paper, we assume that q ≡ 1 mod m , where m is a positive integer. We first study a class of q - ary quantum codes of length r (q m - 1) q - 1 by CSS construction, where r ∣ q - 1 . Then, we construct two families of q - ary quantum codes of length r (q 2 m - 1) q 2 - 1 by Hermitian construction. These constructions produce new quantum codes with larger dimensions. [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, by investigating the solutions of certain equations over finite fields, we make progress towards three conjectures about optimal ternary cyclic codes which were proposed by Ding and Helleseth. [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, two classes of cyclic codes whose duals have two zeros are presented. The weight distributions of these cyclic codes are settled with the help of Gaussian periods. The duals of one class of cyclic codes are also studied. Some of the cyclic codes presented in this paper are optimal in the sense that they meet some bounds of linear codes. [ABSTRACT FROM AUTHOR]

Low density parity check code (LDPC) is a linear block code whose performance is close to Shannon's limit. Compared with other decoding algorithms, it has low decoding complexity and flexible structure. So it becomes people's research hotspots. In recent years, LDPC code codec algorithm has been improved, and is widely used in deep space communications, optical fiber communications, mobile communications and underwater and other fields. Using chaotic sequence single-layer mapping plus pixel scrambling can complete chaotic image encryption, because some chaotic sequence is poor robust, in the process of channel transmission, it is difficult to ensure that the information can be destroyed after the attack In this paper, LDPC codes are introduced, and in the chaotic image encryption, block coding is used to form error protection to improve the stability of the image transmission process and improve the transmission quality. [ABSTRACT FROM AUTHOR]

Assuming that we have a soft-decision list decoding algorithm of a linear code, a new hard-decision list decoding algorithm of its repeated code is proposed in this article. Although repeated codes are not used for encoding data, due to their parameters, we show that they have a good performance with this algorithm. We compare, by computer simulations, our algorithm for the repeated code of a Reed-Solomon code against a decoding algorithm of a Reed-Solomon code. Finally, we estimate the decoding capability of the algorithm for Reed-Solomon codes and show that performance is somewhat better than our estimates. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting class of linear codes due to their efficient encoding and decoding algorithms. Bose–Chaudhuri–Hocquenghem (BCH) codes which form a significant subclass of cyclic codes are important in both theory and practice since they have good error-correcting capabilities and have been widely used in communication systems, storage devices, and so on. Quantum codes with good parameters can be also constructed from BCH codes. In this paper, we construct q-ary quantum codes of length q 2 m - 1 ρ using constacyclic BCH codes with order ρ and cyclic BCH codes, respectively, where ρ divides q + 1 , q is a prime power and m is a positive integer. By comparing the obtained quantum codes, we get that constacyclic BCH codes are a better resource in constructing quantum codes than cyclic BCH codes in general. Compared with the quantum codes available in Aly et al. (IEEE Trans Inf Theory 53(3): 1183–1188, 2007) and Zhang et al. (IEEE Access 4:36122, 2018), the quantum codes in our schemes have better parameters. In particular, we extend some known results in Kai et al. (Int J Quantum Inf 16(7):1850059, 2018), La Guardia (Phys Rev A 80(4):042331, 2009), Li et al. (Quantum Inf Comput 12:0021–0035, 2013), Lin (IEEE Trans Inf Theory 50(3):5551–5554, 2004), Tang et al. (IEICE Trans Fund E102-A(1):303–306, 2019), Wang and Zhu (Quantum Inf Process 14(3):881–889, 2015), Yuan et al. (Des Codes Cryptogr 85(1):179–190, 2017) to more general case. [ABSTRACT FROM AUTHOR]

Some methods have been proposed to construct quantum codes out of classical codes. There is a relationship between quantum codes and codes on the quaternary field. The structure of CSS codes has been raised to construct quantum codes. Most of them are based on the fact that a classical code is self-orthogonal. In this paper, a quantum code is constructed using a suitable permutation-based technique on the parity check matrix of classical codes. Several examples of the new quantum error correcting codes are provided. Single parity check (SPC) product codes have simple decoding algorithms, as well as good decoding efficiency. With this technique, the SPC product codes are used to construct quantum codes. [ABSTRACT FROM AUTHOR]