Stopping Set Distributions of Some Linear Codes

Stopping sets and stopping set distribution of an low-density parity-check code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let $C$ be a binary $[n,k]$ linear code with parity-check matrix $H$, where the rows of $H$ may be dependent. A stopping set $S$ of $C$ with parity-check matrix $H$ is a subset of column indices of $H$ such that the restriction of $H$ to $S$ does not contain a row of weight one. The stopping set distribution $\{T_i(H)\}_{i=0}^n$ enumerates the number of stopping sets with size $i$ of $C$ with parity-check matrix $H$. Note that stopping sets and stopping set distribution are related to the parity-check matrix $H$ of $C$. Let $H^{*}$ be the parity-check matrix of $C$ which is formed by all the non-zero codewords of its dual code $C^{\perp}$. A parity-check matrix $H$ is called BEC-optimal if $T_i(H)=T_i(H^*), i=0,1,..., n$ and $H$ has the smallest number of rows. On the BEC, iterative decoder of $C$ with BEC-optimal parity-check matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes and the extended Hamming codes.Comment: 33 pages, submitted to IEEE Trans. Inform. Theory, Feb. 201

Deterministic Constructions of Binary Measurement Matrices from Finite Geometry

Deterministic constructions of measurement matrices in compressed sensing (CS) are considered in this paper. The constructions are inspired by the recent discovery of Dimakis, Smarandache and Vontobel which says that parity-check matrices of good low-density parity-check (LDPC) codes can be used as {provably} good measurement matrices for compressed sensing under $\ell_1$-minimization. The performance of the proposed binary measurement matrices is mainly theoretically analyzed with the help of the analyzing methods and results from (finite geometry) LDPC codes. Particularly, several lower bounds of the spark (i.e., the smallest number of columns that are linearly dependent, which totally characterizes the recovery performance of $\ell_0$-minimization) of general binary matrices and finite geometry matrices are obtained and they improve the previously known results in most cases. Simulation results show that the proposed matrices perform comparably to, sometimes even better than, the corresponding Gaussian random matrices. Moreover, the proposed matrices are sparse, binary, and most of them have cyclic or quasi-cyclic structure, which will make the hardware realization convenient and easy.Comment: 12 pages, 11 figure

A STUDY OF SELECTED FACTORS AFFECTING TAKEOFF HEIGHT IN THREE-METER SPRINGBOARD DIVING

The purpose of this study was to find out the connection between some factors in the takeoff process and the takeoff height. A Human-Springboard system, based on a previously developed theory and numerical method, was developed to simulate the takeoff process. In addition, this system could output the takeoff height and other results in response to the input of the control function. Through changing the parameters in a certain form of control function, the relationship between those factors and the takeoff height could be determined. The results of this study could be used as a theoretic base for the coaches and athletes

Untargeted Backdoor Watermark: Towards Harmless and Stealthy Dataset Copyright Protection

Deep neural networks (DNNs) have demonstrated their superiority in practice. Arguably, the rapid development of DNNs is largely benefited from high-quality (open-sourced) datasets, based on which researchers and developers can easily evaluate and improve their learning methods. Since the data collection is usually time-consuming or even expensive, how to protect their copyrights is of great significance and worth further exploration. In this paper, we revisit dataset ownership verification. We find that existing verification methods introduced new security risks in DNNs trained on the protected dataset, due to the targeted nature of poison-only backdoor watermarks. To alleviate this problem, in this work, we explore the untargeted backdoor watermarking scheme, where the abnormal model behaviors are not deterministic. Specifically, we introduce two dispersibilities and prove their correlation, based on which we design the untargeted backdoor watermark under both poisoned-label and clean-label settings. We also discuss how to use the proposed untargeted backdoor watermark for dataset ownership verification. Experiments on benchmark datasets verify the effectiveness of our methods and their resistance to existing backdoor defenses. Our codes are available at \url{https://github.com/THUYimingLi/Untargeted_Backdoor_Watermark}.Comment: This work is accepted by the NeurIPS 2022 (Oral, TOP 2%). The first two authors contributed equally to this work. 25 pages. We have fixed some typos in the previous versio

Poly[[[silver(I)-μ-1,4-bis­[(imidazol-1-yl)meth­yl]benzene-κ2 N 3:N 3′-silver(I)-μ-1,4-bis­[(imidazol-1-yl)meth­yl]benzene-κ2 N 3:N 3′] 4,4′-diazenediyldibenzoate] dihydrate]

In the title compound, [Ag2(C14H14N4)2](C14H8N2O4)·2H2O, each of the two unique Ag+ ions is two-coordinated by two N atoms from two different 1,4-bis­[(imidazol-1-yl)meth­yl]benzene ligands in an almost linear fashion [N—Ag—N = 170.34 (10) and 160.25 (10)°]. The 4,4′-diazenediyldibenzoate anions do not coordinate to Ag. O—H⋯O hydrogen bonds stabilize the crystal structure