Visualizing academic lineage is valuable for understanding the transmission of knowledge and the development of disciplines over time.We present Mentorship Navigator, an interactive visualization tool that allows users to explore academic lineages shaped through mentorship.Our tool leverages data from the Math Genealogy Project to examine mentor-mentee relationships dynamically.It also incorporates data from OpenAlex to explore the academic contributions of thesis advisors and their descendants.By supporting interactive and incremental construction of a scholar's genealogy, Mentorship Navigator provides a nuanced view of scholarly trajectories, contributing to a better understanding of their influence on contemporary academic thought.

A large body of work has sought to create effective ways of communicating and interacting with visual information. This paper presents insights into this research landscape based on the proceedings from the International Symposium on Visual Information Communication and Interaction (VINCI). We analyze several aspects of VINCI’s scholarship, including key contributors, emerging topics, and collaboration and citation patterns. Our findings reveal the dynamic character of VINCI and its advancements, highlighting its interdisciplinary nature with contributions from diverse institutions and countries, with the majority of authors engaging in both domestic and international collaborations. The insights from this study can inform future research directions and support more fruitful collaborations among VINCI researchers.

Relational datasets capture insights into complex networks of entities and their interconnections. The analysis of such datasets is critical for various domains, from social and biological networks to scientific research. The complexity and interdependencies inherent to relational datasets present significant challenges for analysts aiming to explore and understand such data. These challenges are particularly notable for individuals lacking expertise in data visualization tools and techniques, as well as those without training in deriving complex relations between the entities contained within the dataset. In this paper, we present a visualization tool designed to facilitate this kind of exploration. We illustrate the use of our tool with a dataset on the scientific production of the VINCI symposium.

Appropriate communication and understanding of political data are key to achieve a healthy dialogue between civil society and political institutions. This is particularly important for data generated by bodies that most directly represent citizens, such as congresses, senates, parliaments, and other forms of legislatures. Visualization tools have the potential to support the exploration and understanding of such data and make it more accessible and appealing to the general public. With this vision, we present LegisLatio, an interactive visualization tool that enables open-ended analysis of legislative roll-call vote data from multi-party electoral systems. We describe the design of the tool and illustrate its effectiveness with data generated by the Ecuadorian legislature. We also discuss the findings of a qualitative observation that hints at LegisLatio’s potential to promote citizen engagement and participation.

We present an interactive visualization tool that enables exploration and comparative analyses of election data in multi-partisan systems. We motivate and explain our design in the context of the Ecuadorian political landscape. We demonstrate the tool with data from Ecuador’s three most recent presidential elections. Our tool enables both relative and absolute comparisons of the election results.

Linear complexity is a measure of how complex a one dimensional sequence can be. In this paper we extend the concept of linear complexity to multiple dimensions and present a definition that is invariant under well-orderings of the arrays. As a result we find that our new definition for the process introduced in the patent titled “Digital Watermarking” produces arrays with good asymptotic properties.

In this paper, we present two generators for the group of symmetries of the generic (m+1) -dimensional periodic Costas arrays over elementary abelian (\BBZp)mgroups: one that is defined by multiplication onmdimensions and the other by shear (addition) onmdimensions. Through exhaustive search, we observe that these two generators characterize the group of symmetries for the examples we were able to compute. Following the results, we conjecture that these generators characterize the group of symmetries of the generic (m+1) -dimensional periodic Costas arrays over elementary abelian (\BBZp)mgroups.

This paper presents two new classes of families of multi-periodic arrays suitable for digital watermarking. These families are large and all arrays have good auto and cross-correlation, and are available in many sizes. The construction is based on log quadratic and exponential quadratic maps. The maps applied to elements in an extension field on a natural multi-dimensional grid. The grid also leads to a generalized multi-dimensional Legendre array, which can be used in the family constructions.

New families of three-dimensional (3-D) optical orthogonal codes for applications to optical code-division multiple access (OCDMA) networks are proposed. The families are based in the three dimensional periodic Welch Costas array over elementary Abelian groups. These new families are shown to be asymptotically optimal through the Johnson bound.

The new generations of cellular technologies and the current trend towards small cells or femtocells, will offer an improved spectral efficiency per area, but they also offer an opportunity to improve the energy efficiency, measured as the power consumption needed to provide a certain throughput in a given area. In this paper, two network performance indicators, aggregated throughput and energy efficiency, have been analyzed and compared in two network architectures; a traditional deployment based on outdoor macro base stations, for the provision of outdoor and indoor coverage, and a deployment where some of the indoor traffic is supported by femtocells. Performance simulations and energy calculations have shown that the introduction of a femtocell layer, complementing the macrocell layer is a great leap forward in system performance and energy efficiency, when compared with current indoor coverage based on outdoor macro and micro base stations. Additionally, this paper present a study on how much could be reduced the mobile network energy requirements of macrocell deployments in high density urban deployments, in low traffic conditions, based on the application of two Self Organizing Networks (SON) techniques for energy saving, selective disconnection and power reduction of eNodeBs and HeNBs.

Digital fingerprinting is a technique for tracing the distribution of multimedia content, and protecting it from unauthorized manipulation. Unique identification information is embedded into each distributed copy of the signal. In a collusion attack, fingerprints are combined to remove or distort the fingerprints. Audio signals are good candidates for fingerprinting, because of the forgiving nature of the human auditory system to cross-talk between channels. We use principal components of the audio signal to construct an abstract vector space. The fingerprints are ordered rotations in that space. The rotations are determined by arrays with good correlation properties. These arrays are embedded in real audio, and are imperceptible, according to a panel of experts. These fingerprints are resistant to an averaging collusion attack by hundreds or thousands of colluders, and can withstand a worst case RandNeg attack by up to 30 colluders.

Improvements to the Johnson Bound for Optical Orthogonal Codes have been used to prove the optimality of double-periodic arrays with row and column length relatively prime. In our work we produce families of double-periodic arrays where the row and column length are not relatively prime. In this work we introduce the concept of group permutable constant weight codes and non-binary group permutable constant weight codes. We present improvements to the Johnson Bound to bound the cardinality of the families of double-periodic arrays whose row and column length are not relatively prime. We present some families of group permutable constant weight codes and prove the optimality of these families.

This paper presents new families of two and three-dimensional arrays designed for watermarking. The array construction is algebraic, based on finite fields. The arrays have good auto and cross-correlation with other family members. The family size is large enough to be used in watermarking at the point of origin, providing a unique watermark for every video and image. Such watermarks have applications in surveillance, sub-image object tracking and as fingerprints for proof of tampering, audit trail monitoring and traitor tracing.

Yang and Fuja presented constructions of codes with unequal correlation constraints (lambdaca). And specifically they have constructions for the case where lambdaa= 2 > lambdac= 1. In their work they argue that it is important to make the cross-correlation (lambdac) as small as possible, and not necessarily lambdaa= lambdac. In this work we present a method to generate new Yang-Fuja type families with correlation constraints where lambdaca. We presented a method to increase the size of a family of a double periodic sequence with optical orthogonal code (OOC) length n = mp where m = p - 1, and p is a prime. In this work we present a new method that increases the size of a family of double periodic sequences without the restriction on m, we present a method to increase the weight of a double periodic array, and then combine both methods to produce Yang-Fuja type families of double periodic arrays with lambdaca. One of our main results is a theorem (Theorem 3, Section IV) that estimates lambdac. With this theorem we obtain two new constructions of optical orthogonal codes with lambdaca.

A new construction of optical orthogonal codes is provided in this correspondence which is a generalization of the well-known construction of distinct difference set (DDS) by Bose and Chowla. This construction is optimal with respect to the Johnson bound and has parameters$n=q^a-1,$$\omega=q,$and$\lambda=1$.

In this paper we present an algorithmic approach to the problem of the divisibility of the number of solutions to a system of polynomial equations. Using this method we prove that all binary cyclic codes with two zeros over F2fand minimum distance 5 are quasi-perfect for f les 10. We also present elementary proofs of divisibility results that, in some cases, improve previous results

The cyclic difference sets constructed by Singer are also examples of perfect distinct difference sets (DDS). The Bose construction of distinct difference sets, leads to a relative difference set. In this paper we introduce the concept of partial relative DDS and prove that an optical orthogonal code (OOC) construction due to Moreno et. al., is a partial relative DDS. We generalize the concept of ideal matrices previously introduced by Kumar and relate it to the concepts of this paper. Another variation of ideal matrices is introduced in this paper: Welch ideal matrices of dimension n by (n - 1). We prove that Welch ideal matrices exist only for n prime. Finally, we recast an old conjecture of Golomb on the Welch construction of Costas arrays using the concepts of this paper. This connection suggests that our construction of partial relative difference sets is in a sense, unique

There are only a few multiple target families of Costas and sonar arrays with perfect correlation property. In this paper using the Welch Costas array and some results from design theory we construct perfect auto and cross-correlation families of sonar and extended Costas arrays

In this correspondence, we correct an error in the correspondence cited in the title.

Optical orthogonal codes (OOC) are used as spreading sequences for optical CDMA networks. An OOC is a family of constant weight binary codes with a pre-specified maximum correlation parameter (MCP). Johnson in his 1962 paper introduced three bounds for constant weight codes, that we call bounds A, B, and hybrid. Subsequently Chung et al. adapted Johnson bound A to generate a bound for OOCs, which has been widely used to prove the optimality of OOCs. Johnson bound B has been used in a prior work of this paper's authors to prove the optimality of some OOCs. In this paper we give an improvement of this bound, and based on that prove the optimality of some other constructions which were not known to be optimal. Using the results from Agrell et al., 2000 paper we also give an improvement of Johnson hybrid bound for constant weight codes, and then use it to generate a bound for OOCs. Finally, we introduce a new family of OOCs, based on flats in an affine geometry. While OOCs based on lines and hyperplanes are optimal, we can't say much about other OOCs resulting from this construction. We show that the hybrid bound gives tighter bound than the other two bounds in some regions for this construction. Recently lot of interest has been shown to find all optimal OOCs with weight 4 and 5 and MCP 1 and 2. Using affine geometry construction, a new family of optimal OOCs with weight 4 and MCP 2 is introduced

This paper describes an application of our generalization of McEliece's theorem in order to improve Serre's bound. Serre obtained sharp estimates for the number of solutions of an algebraic curve over a finite field. This improvement depends of the divisibility properties of the curve. Curves with many rational points are important in algebraic-geometry codes and low-discrepancy sequences

This paper presents a new optimal double periodical construction of optical orthogonal codes for multiple and two-dimensional double-periodic arrays with auto and cross correlation constraints. The new construction is optimal in the Johnson bound to obtain an optimal family of extended sonar type arrays with the property of double periodicity. The equivalence theorem used to obtain the signal pattern provides families of multiple target arrays.

Two new optimal constructions of optical orthogonal codes with lambdages2 are introduced. The first is based on a previous construction for the case lambda=1. The second is based on difference sets. A new bound for optical orthogonal codes based on a known bound for constant weight codes is introduced. This bound is used to prove the optimality of our constructions

We are presenting a new method to obtain the covering radius of codes and in particular to prove quasi-perfection in codes. Our techniques combine divisibility results of Ax-Katz and Moreno-Moreno as well as coding theoretic methods. We answer a problem posed by Cohen-Honkala-Litsyn-Lobstein in the book covering radius for Bose-Chaudhuri-Hocquenghem (BCH) codes. We also obtain the covering radius for many new classes of codes.

We give an algebraic construction for a new family of frequency-hop codes. The construction is based on properties of finite fields: it is shown that for each field GF(p/sup m/), there exists a large number of codes of length p/sup m/. The codes are also shown to possess the best possible simultaneous two-dimensional autocorrelation and cross-correlation properties. Moreover, they include a family of codes: with a code length of a power of 2, which are ideally suitable for applications in digital communication systems.

We consider certain subcodes of generalized Reed-Muller (GRM) codes, which we call homogeneous generalized Reed-Muller (HRM) codes. In general, they have a much better minimum distance than the GRM codes. The parameters of HRM codes are related to those of projective Reed-Muller (PRM) codes. Unlike most PRM codes, punctured HRM codes are cyclic. Under the trace map, HRM codes map to binary codes. These are in general much larger than classical RM codes, for the same minimum distance.

An improvement on the Serre bound for the number of rational places is obtained. Using our result we improve the bounds on the generalized Hamming weights of the duals of the BCH codes estimated by Stichtenoth and Voss (1994). Our new bound is tight in the binary case of the generalized Hamming weight of BCH (2)/sup /spl perp//.

Sonar sequences were introduced by Golomb and Taylor in 1982. We introduce the concept of extended sonar sequences, which is similar to that of sonar sequences except that blank columns are permitted. Several constructions for extended sonar sequences are offered here. Some of these are close to constructions for ordinary sonar sequences, but they provide improvements to the list of best sonar sequences with up to 100 symbols.

Golomb and Taylor (1984) conjectured that single periodicity characterizes the Welch construction of Costas arrays. In this correspondence, we present a weakened version of this conjecture and partial results on it. Furthermore, we prove that our conjecture is equivalent to a conjecture concerning permutation polynomials and give partial results on the latter.

An upper hound for Weil-type exponential sums over Galois rings was derived by Kumar, Helleseth, and Calderbank (see ibid., vol.41, no.3, p.456, 1995). This bound leads directly to an estimate for the minimum distance of Z/sub 4/-linear trace codes. An improved minimum-distance estimate is presented. First, McEliece's result on the divisibility of the weights of binary cyclic codes is extended to Z/sub 4/ trace codes. The divisibility result is then combined with the techniques of Serre (1983) and of Moreno and Moreno (see ibid., vol.40, no.11, p.1101, 1994) to derive the improved minimum-distance estimate. The improved estimate is tight for the Kerdock code as well as for the Delsarte-Goethals codes.

Three new constructions for families of cyclic constant weight codes are presented. All are asymptotically optimum in the sense that in each case, as the length of the sequences within the family approaches infinity, the ratio of family size to the maximum possible under the Johnson upper bound, approaches unity.>

Using known techniques of desingularization for singular plane projective curves over finite fields F/sub q/, q=2/sup m/, we found several new binary plane projective curves of genus 3, 4, and 5 with the maximal number of F/sub q/-rational points (q=2/sup m/, m=3, 4, 5, 6, 7, 8, and 9) on their smooth projective models, which are close to or meet Serre's upper bound.

Research Problem 9.5 of MacWilliams and Sloane's book, The Theory of Error Correcting Codes (Amsterdam: North-Holland, 1977), asks for an improvement of the minimum distance bound of the duals of BCH codes, defined over F/sub 2/m with m odd. The objective of the present article is to give a solution to the above problem by with (i) obtaining an improvement to the Ax (1964) theorem, which we prove is the best possible for many classes of examples; (ii) establishing a sharp estimate for the relevant exponential sums, which implies a very good improvement for the minimum distance bounds; (iii) providing a doubly infinite family of counterexamples to Problem 9.5 where both the designed distance and the length increase independently; (iv) verifying that our bound is tight for some of the counterexamples; and (v) in the case of even m, giving a doubly infinite family of examples where the Carlitz-Uchiyama bound is tight, and in this way determining the exact minimum distance of the duals of the corresponding BCH codes.>

The authors construct new and improved sonar sequences by applying rotation, multiplication, and shearing transformations to Costas sequence constructions. A catalog of the best known sonar sequences with up to 100 symbols is given.>

At the present time, there are very good methods to obtain bounds for the minimum distance of BCH codes and their duals. On the other hand, there are few other bounds suitable for general cyclic codes. Therefore, research Problem 9.9 of MacWilliams and Sloane (1977), The Theory of Error-Correcting Codes, asks if the bound of Deligne (1974) for exponential sums in several variables or the bound of Lang and Weil (1954), can be used to obtain bounds on the minimum distance of codes. This question is answered in the affirmative by showing how Deligne's theorem can be made to yield a lower bound on the minimum distance of certain classes of cyclic codes. In the process, an infinite family of binary cyclic codes is presented for which the bound on minimum distance so derived is as tight as possible. In addition, an infinite family of polynomials of degree 3 in 2 variables over a field of characteristic 2, for which Deligne's bound is tight, is exhibited. Finally, a bound is presented for the minimum distance of the duals of the binary subfield subcodes of generalized Reed-Muller codes as well as for the corresponding cyclic codes. It is noted that these codes contain examples of the best binary cyclic codes.>

For pt.I, see Proc. AMS, vol.III, p.523-31 (1991). The minimum distance of a Goppa code is found when the length of code satisfies a certain inequality on the degree of the Goppa polynomial. In order to do this, conditions are improved on a theorem of E. Bombieri (1966). This improvement is used also to generalize a previous result on the minimum distance of the dual of a Goppa code. This approach is generalized and results are obtained about the parameters of a class of subfield subcodes of geometric Goppa codes; in other words, the covering radii are estimated, and further, the number of information symbols whenever the minimum distance is small in relation to the length of the code is found. Finally, a bound on the minimum distance of the dual code is discussed.>

For the case where p is an odd prime, n>or=2 is an integer, and omega is a complex primitive pth root of unity, a construction is presented for a family of p/sup n/ p-phase sequences (symbols of the form omega /sup i/), where each sequence has length p/sup n/-1, and where the maximum nontrivial correlation value C/sub max/ does not exceed 1+ square root p/sup n/. A complete distribution of correlation values is provided. As a special case of this construction, a previous construction due to Sidelnikov (1971) is obtained. The family of sequences is asymptotically optimum with respect to its correlation properties, and, in comparison with many previous nonbinary designs, the present design has the additional advantage of not requiring an alphabet of size larger than three. The new sequences are suitable for achieving code-division multiple access and are easily implemented using shift registers. They wee discovered through an application of Deligne's bound (1974) on exponential sums of the Weil type in, several variables. The sequences are also shown to have strong identification with certain bent functions.>

It is known that extended Goppa cedes are invariant under the group of transformationsZ \rightarrow (A Z + B ) / ( CZ + D ), withA D + BC \neq 0. This invariance is used here to classify cubic and quartic irreducible Goppa codes and to investigate their symmetry groups. A computer has been used to determine the actual group of the codes of length 33 (for cubics and quarries). It has been said, concerning the trends in symmetry groups with respect to the Gilbert bound, that "a good family of codes can be linear or have many symmetries, hut not both" [8]. The groups found here are rather small; and so the results reinforce that statement.

The class of codes introduced by Goppa [1]-[3] includes the BCH codes as a proper subset. It also includes a large subset of asymptotically good codes, each of which has an algebraic decoding algorithm for correcting some smaller number of errors. In Section 7 of [1], Goppa gives necessary and sufficient conditions for his codes to be isomorphic to cyclic codes under a certain correspondence. In this correspondence, we exhibit another correspondence which reveals that certain other Goppa codes (including the example of Goppa's Section 6) become cyclic when extended by an overall parity check. In particular, the extended Goppa codes with(n,k,d) = (2^m + 1, 2^m - 2m, 6)are isomorphic to the reversible cyclic codes with check polynomial(x + 1)f(x), wheref(x)is an irreducible polynomial of period2^m + 1.