Rectangular, range, and restricted AONTs: Three generalizations of all-or-nothing transforms

<p style='text-indent:20px;'>All-or-nothing transforms (AONTs) were originally defined by Rivest [<xref ref-type="bibr" rid="b14">14</xref>] as bijections from <inline-formula><tex-math id="M1">\begin{document}$ s $\end{document}</tex-math></inline-formula> input blocks to <inline-formula><tex-math id="M2">\begin{document}$ s $\end{document}</tex-math></inline-formula> output blocks such that no information can be obtained about any input block in the absence of any output block. Numerous generalizations and extensions of all-or-nothing transforms have been discussed in recent years, many of which are motivated by diverse applications in cryptography, information security, secure distributed storage, etc. In particular, <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-AONTs, in which no information can be obtained about any <inline-formula><tex-math id="M4">\begin{document}$ t $\end{document}</tex-math></inline-formula> input blocks in the absence of any <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> output blocks, have received considerable study.</p><p style='text-indent:20px;'>In this paper, we study three generalizations of AONTs that are motivated by applications due to Pham et al. [<xref ref-type="bibr" rid="b13">13</xref>] and Oliveira et al. [<xref ref-type="bibr" rid="b12">12</xref>]. We term these generalizations rectangular, range, and restricted AONTs. Briefly, in a rectangular AONT, the number of outputs is greater than the number of inputs. A range AONT satisfies the <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula>-AONT property for a range of consecutive values of <inline-formula><tex-math id="M7">\begin{document}$ t $\end{document}</tex-math></inline-formula>. Finally, in a restricted AONT, the unknown outputs are assumed to occur within a specified set of "secure" output blocks. We study existence and non-existence and provide examples and constructions for these generalizations. We also demonstrate interesting connections with combinatorial structures such as orthogonal arrays, split orthogonal arrays, MDS codes and difference matrices.</p>

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever $$v \equiv 39$$ v ≡ 39 (mod 72), or $$v \equiv 4^e48 + 3$$ v ≡ 4 e 48 + 3 (mod $$4^e96$$ 4 e 96 ) and $$e \ge 0$$ e ≥ 0 , there exists a KTS on v points having at least $$v-3$$ v - 3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group G acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of G, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.

Convolution Neural Networks for Localization of Near-Field Sources via Symmetric Double-Nested Array

We present the convolution neural networks (CNNs) to achieve the localization of near-field sources via the symmetric double-nested array (SDNA). Considering that the incoherent near-field sources can be separated in the frequency spectrum, we first calculate the phase difference matrices and consider the typical elements as the inputs of the networks. In order to guarantee the precision of the angle-of-arrival (AOA) estimation, we implement the autoencoders to divide the AOA subregions and construct the corresponding classification CNNs to obtain the AOAs of near-field sources. Then, we construct a particular range vector without the estimated AOAs and utilize the regression CNN to obtain the range parameters of near-field sources. The proposed algorithm is robust to the off-grid parameters and suitable for the scenarios with the different number of near-field sources. Moreover, the proposed method outperforms the existing method for near-field source localization.

Constructions of irredundant orthogonal arrays

<p style='text-indent:20px;'>An <inline-formula><tex-math id="M1">\begin{document}$ N \times k $\end{document}</tex-math></inline-formula> array <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math></inline-formula> with entries from <inline-formula><tex-math id="M3">\begin{document}$ v $\end{document}</tex-math></inline-formula>-set <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{V} $\end{document}</tex-math></inline-formula> is said to be an <i>orthogonal array</i> with <inline-formula><tex-math id="M5">\begin{document}$ v $\end{document}</tex-math></inline-formula> levels, strength <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula> and index <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, denoted by OA<inline-formula><tex-math id="M8">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula>, if every <inline-formula><tex-math id="M9">\begin{document}$ N\times t $\end{document}</tex-math></inline-formula> sub-array of <inline-formula><tex-math id="M10">\begin{document}$ A $\end{document}</tex-math></inline-formula> contains each <inline-formula><tex-math id="M11">\begin{document}$ t $\end{document}</tex-math></inline-formula>-tuple based on <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{V} $\end{document}</tex-math></inline-formula> exactly <inline-formula><tex-math id="M13">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> times as a row. An OA<inline-formula><tex-math id="M14">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula> is called <i>irredundant</i>, denoted by IrOA<inline-formula><tex-math id="M15">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula>, if in any <inline-formula><tex-math id="M16">\begin{document}$ N\times (k-t ) $\end{document}</tex-math></inline-formula> sub-array, all of its rows are different. Goyeneche and <inline-formula><tex-math id="M17">\begin{document}$ \dot{Z} $\end{document}</tex-math></inline-formula>yczkowski firstly introduced the definition of an IrOA and showed that an IrOA<inline-formula><tex-math id="M18">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula> corresponds to a <inline-formula><tex-math id="M19">\begin{document}$ t $\end{document}</tex-math></inline-formula>-uniform state of <inline-formula><tex-math id="M20">\begin{document}$ k $\end{document}</tex-math></inline-formula> subsystems with local dimension <inline-formula><tex-math id="M21">\begin{document}$ v $\end{document}</tex-math></inline-formula> (Physical Review A. 90 (2014), 022316). In this paper, we present some new constructions of irredundant orthogonal arrays by using difference matrices and some special matrices over finite fields, respectively, as a consequence, many infinite families of irredundant orthogonal arrays are obtained. Furthermore, several infinite classes of <inline-formula><tex-math id="M22">\begin{document}$ t $\end{document}</tex-math></inline-formula>-uniform states arise from these irredundant orthogonal arrays.</p>

Discrete Correlation of Order 2 of Generalized Rudin-Shapiro Sequences on Alphabets of Arbitrary Size

In 2009, Grant, Shallit, and Stoll [Acta Arith. 140 (2009), [345–368] constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2 in cases where the size of the alphabet is a prime number or a squarefree product of primes. We establish similar results for an even larger family of pseudorandom sequences, constructed via difference matrices, in the case of an alphabet of any size. The constructions generalize those from Grant et al. In the case where the size of the alphabet is squarefree and where there are at least two prime factors, we obtain an improvement in the error term by comparison with the result of Grant et al.

Device-Free Wireless Localization Using Artificial Neural Networks in Wireless Sensor Networks

Currently, localization has been one of the research hot spots in Wireless Sensors Networks (WSNs). However, most localization methods focus on the device-based localization, which locates targets with terminal devices. This is not suitable for the application scenarios like the elder monitoring, life detection, and so on. In this paper, we propose a device-free wireless localization system using Artificial Neural Networks (ANNs). The system consists of two phases. In the off-line training phase, Received Signal Strength (RSS) difference matrices between the RSS matrices collected when the monitoring area is vacant and with a professional in the area are calculated. Some RSS difference values in the RSS difference matrices are selected. The RSS difference values and corresponding matrix indices are taken as the inputs of an ANN model and the known location coordinates are its outputs. Then a nonlinear function between the inputs and outputs can be approximated through training the ANN model. In the on-line localization phase, when a target is in the monitoring area, the RSS difference values and their matrix indices can be obtained and input into the trained ANN model, and then the localization coordinates can be computed. We verify the proposed device-free localization system with a WSN platform. The experimental results show that our proposed device-free wireless localization system is able to achieve a comparable localization performance without any terminal device.