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In 2018, J. M. Gómez et al. showed that the problem of finding the packing number $ \rho(F_2(P_n)) $ of the 2-token graph $ F_2(P_n) $ of the path $ P_n $ of length $ n\ge 2 $ is equivalent to determining the maximum size of a binary code $ S' $ of constant weight $ w = 2 $ that can correct a single adjacent transposition. By determining the exact value of $ \rho(F_2(P_n)) $, they proved a conjecture of Rob Pratt. In this paper, we study a related problem, which consists of determining the packing number $ \rho(F_3(P_n)) $ of the graph $ F_3(P_n) $. This problem corresponds to the Sloane's problem of finding the maximum size of $ S' $ of constant weight $ w = 3 $ that can correct a single adjacent transposition. Since the maximum packing set problem is computationally equivalent to the maximum independent set problem, which is an NP-hard problem, then no polynomial time algorithms are expected to be found. Nevertheless, we compute the exact value of $ \rho(F_3(P_n)) $ for $ n\leq 12 $, and we also present some algorithms that produce a lower bound for $ \rho(F_3(P_n)) $ with $ 13\leq n\leq 44 $. Finally, we establish an upper bound for $ \rho(F_3(P_n)) $ with $ n\geq 13 $.

Abstract A novel channel code, the low‐density algebra‐check (LDAC) code, is introduced in this letter. LDAC code is constructed by combining sparse matrice with algebraic constraints from two types of subcodes. Detailed explanations for the encoding and decoding process of the LDAC codes are provided in the paper. Simulation results demonstrate that LDAC codes offer advantages over LDPC codes in constructing low rate codes. The proposed ultra‐low rate LDAC code can achieve approximately 0.4 and 0.2 dB improvement in SNR compared to CA‐polar and 5G LDPC, respectively, at the 10−6 block error rate (BER) over the additive white Gaussian noise (AWGN) channel. LDAC codes demonstrate superior performance over LDPC codes at low rates, due to the enhanced error correction capabilities of the two types of subcodes and reduced rates without introducing short cycles.

In visual indoor positioning systems, the method of constructing a visual map by point-by-point sampling is widely used due to its characteristics of clear static images and simple coordinate calculation. However, too small a sampling interval will cause image redundancy, while too large a sampling interval will lead to the absence of any scene images, which will result in worse positioning efficiency and inferior positioning accuracy. As a result, this paper proposed a visual map construction method based on pre-sampled image features matching, according to the epipolar geometry of adjacent position images, to determine the optimal sampling spacing within the constraints and effectively control the database size while ensuring the integrity of the image information. In addition, in order to realize the rapid retrieval of the visual map and reduce the positioning error caused by the time overhead, an image retrieval method based on deep hashing was also designed in this paper. This method used a convolutional neural network to extract image features to construct the semantic similarity structure to guide the generation of hash code. Based on the log-cosh function, this paper proposed a loss function whose function curve was smooth and not affected by outliers, and then integrated it into the deep network to optimize parameters, for fast and accurate image retrieval. Experiments on the FLICKR25K dataset and the visual map proved that the method proposed in this paper could achieve sub-second image retrieval with guaranteed accuracy, thereby demonstrating its promising performance.

Background: Gait recognition is perceived as the most promising biometric approach for future decades especially because of its efficient applicability in surveillance systems. Due to recent growth in the use of gait biometrics across surveillance systems, the ability to rapidly search for the required data has become an emerging need. Therefore, we addressed the gait retrieval problem, which retrieves people with gaits similar to a query subject from a large-scale dataset. Methods: This paper presents the deep gait retrieval hashing (DGRH) model to address the gait retrieval problem for large-scale datasets. Our proposed method is based on a supervised hashing method with a deep convolutional network. We use the ability of the convolutional neural network (CNN) to capture the semantic gait features for feature representation and learn the compact hash codes with the compatible hash function. Therefore, our DGRH model combines gait feature learning with binary hash codes. In addition, the learning loss is designed with a classification loss function that learns to preserve similarity and a quantization loss function that controls the quality of the hash codes Results: The proposed method was evaluated against the CASIA-B, OUISIR-LP, and OUISIR-MVLP benchmark datasets and received the promising result for gait retrieval tasks. Conclusions: The end-to-end deep supervised hashing model is able to learn discriminative gait features and is efficient in terms of the storage memory and speed for gait retrieval.