Search

We establish a simple yet general parameterization of Higgs-muon interactions within the effective field theory frameworks, including both the Higgs Effective Field Theory (HEFT) and the Standard Model Effective Field Theory (SMEFT). We investigate the potential of a muon collider, operating at center-of-mass energies of 3 and 10 TeV, to probe Higgs-muon interactions. All possible processes involving the direct production of multiple electroweak bosons ($W$, $Z$, and $H$) with up to five final-state particles are considered. Our findings indicate that a muon collider can achieve greater sensitivity than the high-luminosity LHC, especially considering the independence of the Higgs decay branching fraction to muons. Notably, a 10 TeV muon collider offers exceptional sensitivity to muon-Higgs interactions, surpassing the 3 TeV option. In particular, searches based on multi-Higgs production prove highly effective for probing these couplings., Comment: 6 pages, 4 figures

One method of constructing $(a^2+1, 2,a, 1)$-SEDFs (i.e., strong external difference families) in $\mathbb{Z}_{a^2+1}$ makes use of $\alpha$-valuations of complete bipartite graphs $K_{a,a}$. We explore this approach and we provide a classification theorem which shows that all such $\alpha$-valuations can be constructed recursively via a sequence of ``blow-up'' operations. We also enumerate all $(a^2+1, 2,a, 1)$-SEDFs in $\mathbb{Z}_{a^2+1}$ for $a \leq 14$ and we show that all these SEDFs are equivalent to $\alpha$-valuations via affine transformations. Whether this holds for all $a > 14$ as well is an interesting open problem. We also study SEDFs in dihedral groups, where we show that two known constructions are equivalent.

In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the \emph{nested point}) to every block of a $(v,k,\lambda)$-BIBD in such a way that we end up with a partial $(v,k+1,\lambda+1)$-BIBD. In the case where the partial $(v,k+1,\lambda+1)$-BIBD is in fact a $(v,k+1,\lambda+1)$-BIBD, we have a \emph{perfect nesting}. We show that a nesting is perfect if and only if $k = 2 \lambda + 1$. Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., $(v,3,1)$-BIBDs) when $v \equiv 1 \bmod 6$, as well as for some symmetric BIBDs. Here we study nestings of $(v,4,1)$-BIBDs, which are not perfect nestings. We prove that there is a nested $(v,4,1)$-BIBD if and only if $v \equiv 1 \text{ or } 4 \bmod 12$, $v \geq 13$. This is accomplished by a variety of direct and recursive constructions.

We study the capabilities of a muon collider, at 3 and 10 TeV center-of-mass energy, of probing the interactions of the Higgs boson with the muon. We consider all the possible processes involving the direct production of EW bosons ($W,Z$ and $H$) with up to five particles in the final state. We study these processes both in the HEFT and SMEFT frameworks, assuming that the dominant BSM effects originate from the muon Yukawa sector. Our study shows that a Muon Collider has sensitivity beyond the LHC, as it not only relies on the Higgs-decay branching fraction to muons. A 10 TeV muon collider provides a unique sensitivity on muon and (multi-) Higgs interactions, significantly better than the 3 TeV option. We find searches based purely on multi-Higgs production to be particularly effective in probing these couplings., Comment: Published on JHEP

We establish a first and second-order approximation for an infinite dimensional limit order book model (LOB) in a single (''critical'') scaling regime where market and limit orders arrive at a common time scale. With our choice of scaling we obtain non-degenerate first-order and second-order approximations for the price and volume dynamics. While the first-order approximation is given by a standard coupled ODE-PDE system, the second-order approximation is non-standard and described in terms of an infinite-dimensional stochastic evolution equation driven by a cylindrical Brownian motion. The driving noise processes exhibit a non-trivial correlation in terms of the model parameters. We prove that the evolution equation has a unique solution and that the sequence of standardized LOB models converges weakly to the solution of the evolution equation. The proof uses a non-standard martingale problem. We calibrate a simplified version of our model to market data and show that the model accurately captures correlations between price and volume fluctuations.

A $3$-$(v,\{4,6\},1)$ design is a configuration of $v$ points and a collection of $4$- and $6$-element subsets called blocks, that jointly contain every 3-element subset exactly once. Using an exhaustive computer search on $v\leq 28$ points we investigate the $3$-$(v,\{4,6\},1)$ designs that have a transitive automorphism group and where the blocks of size 6 form a 2-class symmetric design. A 2-class symmetric design with parameters $(v,k{;}\lambda_1,\lambda_2{;}\delta_1,\delta_2)$ is a set-system on $v$ points and $v$ blocks of size $k$, where every pair of points are in $\lambda_1$ or $\lambda_2$ blocks and every pair of blocks intersect in $\delta_1$ or $\delta_2$ points. The 2-class symmetric designs include biplanes, semi-biplanes, and 2-class symmetric partially balanced incomplete block designs.

A muon collider would enable the big jump ahead in energy reach that is needed for a fruitful exploration of fundamental interactions. The challenges of producing muon collisions at high luminosity and 10~TeV centre of mass energy are being investigated by the recently-formed International Muon Collider Collaboration. This Review summarises the status and the recent advances on muon colliders design, physics and detector studies. The aim is to provide a global perspective of the field and to outline directions for future work., Comment: 118 pages, 103 figures

The concept of correlated noise is well-established in discrete-time stochastic modelling but there is no generally agreed-upon definition of the notion of red noise in continuous-time stochastic modelling. Here we discuss the generalization of discrete-time correlated noise to the continuous case. We give an overview of existing continuous-time approaches to model red noise, which relate to their discrete-time analogue via characteristics like the autocovariance structure or the power spectral density. The implications of carrying certain attributes from the discrete-time to the continuous-time setting are explored while assessing the inherent ambiguities in such a generalization. We find that the attribute of a power spectral density decaying as $S(\omega)\sim\omega^{-2}$ commonly ascribed to the notion of red noise has far reaching consequences when posited in the continuous-time stochastic differential setting. In particular, any It\^{o}-differential $\mathrm{d} Y_t=\alpha_t\mathrm{d} t+\beta_t\mathrm{d} W_t$ with continuous, square-integrable integrands must have a vanishing martingale part, i.e. $\mathrm{d} Y_t=\alpha_t\mathrm{d} t$ for almost all $t\geq 0$. We further argue that $\alpha$ should be an Ornstein-Uhlenbeck process.

We explore the sensitivity of directly testing the muon-Higgs coupling at a high-energy muon collider. This is strongly motivated if there exists new physics that is not aligned with the Standard Model Yukawa interactions which are responsible for the fermion mass generation. We illustrate a few such examples for physics beyond the Standard Model. With the accidentally small value of the muon Yukawa coupling and its subtle role in the high-energy production of multiple (vector and Higgs) bosons, we show that it is possible to measure the muon-Higgs coupling to an accuracy of ten percent for a 10 TeV muon collider and a few percent for a 30 TeV machine by utilizing the three boson production, potentially sensitive to a new physics scale about $\Lambda \sim$ 30-100 TeV., Comment: 6 pages, 4 figures, contribution to proceedings of 41st International Conference on High Energy physics - ICHEP2022

The existence of metallic implants in projection images for cone-beam computed tomography (CBCT) introduces undesired artifacts which degrade the quality of reconstructed images. In order to reduce metal artifacts, projection inpainting is an essential step in many metal artifact reduction algorithms. In this work, a hybrid network combining the shift window (Swin) vision transformer (ViT) and a convolutional neural network is proposed as a baseline network for the inpainting task. To incorporate metal information for the Swin ViT-based encoder, metal-conscious self-embedding and neighborhood-embedding methods are investigated. Both methods have improved the performance of the baseline network. Furthermore, by choosing appropriate window size, the model with neighborhood-embedding could achieve the lowest mean absolute error of 0.079 in metal regions and the highest peak signal-to-noise ratio of 42.346 in CBCT projections. At the end, the efficiency of metal-conscious embedding on both simulated and real cadaver CBCT data has been demonstrated, where the inpainting capability of the baseline network has been enhanced.

Non-overlapping codes have been studied for almost 60 years. In such a code, no proper, non-empty prefix of any codeword is a suffix of any codeword. In this paper, we study codes in which overlaps of certain specified sizes are forbidden. We prove some general bounds and we give several constructions in the case of binary codes. Our techniques also allow us to provide an alternative, elementary proof of a lower bound on non-overlapping codes due to Levenshtein in 1964., Comment: 17 pages. Theorems etc renumbered

We develop a cross-border market model between two countries in which the transmission capacities that enable transactions between market participants of different countries are limited. Starting from two reduced-form representations of national limit order book dynamics, we allow incoming market orders to be matched with standing volumes of the foreign market, resulting in cross-border trades. Since the transmission capacities in our model are limited, our model alternates between regimes in which cross-border trades are possible and regimes in which incoming market orders can only be matched against limit orders of the same origin. We derive a high-frequency approximation of our microscopic model, assuming that the size of an individual order converges to zero while the order arrival rate tends to infinity. If transmission capacities are available, the limit process behaves as follows: the volume dynamics is a four-dimensional linear Brownian motion in the positive orthant with oblique reflection at the axes. Each time two queues simultaneously hit zero, the process is reinitialized. The capacity turns out to be a continuous process of finite variation. The analytic tractability of the limiting dynamics allows us to compute key quantities of interest like the distribution of the duration until the next price change. Additionally, we study the effect of cross-border trading on price stability through a simulation study.

In several image acquisition and processing steps of X-ray radiography, knowledge of the existence of metal implants and their exact position is highly beneficial (e.g. dose regulation, image contrast adjustment). Another application which would benefit from an accurate metal segmentation is cone beam computed tomography (CBCT) which is based on 2D X-ray projections. Due to the high attenuation of metals, severe artifacts occur in the 3D X-ray acquisitions. The metal segmentation in CBCT projections usually serves as a prerequisite for metal artifact avoidance and reduction algorithms. Since the generation of high quality clinical training is a constant challenge, this study proposes to generate simulated X-ray images based on CT data sets combined with self-designed computer aided design (CAD) implants and make use of convolutional neural network (CNN) and vision transformer (ViT) for metal segmentation. Model test is performed on accurately labeled X-ray test datasets obtained from specimen scans. The CNN encoder-based network like U-Net has limited performance on cadaver test data with an average dice score below 0.30, while the metal segmentation transformer with dual decoder (MST-DD) shows high robustness and generalization on the segmentation task, with an average dice score of 0.90. Our study indicates that the CAD model-based data generation has high flexibility and could be a way to overcome the problem of shortage in clinical data sampling and labelling. Furthermore, the MST-DD approach generates a more reliable neural network in case of training on simulated data.

In the path towards a muon collider with center of mass energy of 10 TeV or more, a stage at 3 TeV emerges as an appealing option. Reviewing the physics potential of such muon collider is the main purpose of this document. In order to outline the progression of the physics performances across the stages, a few sensitivity projections for higher energy are also presented. There are many opportunities for probing new physics at a 3 TeV muon collider. Some of them are in common with the extensively documented physics case of the CLIC 3 TeV energy stage, and include measuring the Higgs trilinear coupling and testing the possible composite nature of the Higgs boson and of the top quark at the 20 TeV scale. Other opportunities are unique of a 3 TeV muon collider, and stem from the fact that muons are collided rather than electrons. This is exemplified by studying the potential to explore the microscopic origin of the current $g$-2 and $B$-physics anomalies, which are both related with muons., Comment: 73 pages, 28 figures; Contribution to Snowmass 2021

In cone-beam X-ray transmission imaging, perspective deformation causes difficulty in direct, accurate geometric assessments of anatomical structures. In this work, the perspective deformation correction problem is formulated and addressed in a framework using two complementary (180{\deg}) views. The complementary view setting provides a practical way to identify perspectively deformed structures by assessing the deviation between the two views. It also provides bounding information and reduces uncertainty for learning perspective deformation. Two representative networks Pix2pixGAN and TransU-Net for correcting perspective deformation are investigated. Experiments on numerical bead phantom data demonstrate the advantage of complementary views over orthogonal views or a single view. They show that Pix2pixGAN as a fully convolutional network achieves better performance in polar space than Cartesian space, while TransU-Net as a transformer-based hybrid network achieves comparable performance in Cartesian space to polar space. Further study demonstrates that the trained model has certain tolerance to geometric inaccuracy within calibration accuracy. The efficacy of the proposed framework on synthetic projection images from patients' chest and head data as well as real cadaver CBCT projection data and its robustness in the presence of bulky metal implants and surgical screws indicate the promising aspects of future real applications., Comment: 19 pages, 26 figures

The positive outcome of a trauma intervention depends on an intraoperative evaluation of inserted metallic implants. Due to occurring metal artifacts, the quality of this evaluation heavily depends on the performance of so-called Metal Artifact Reduction methods (MAR). The majority of these MAR methods require prior segmentation of the inserted metal objects. Therefore, typically a rather simple thresholding-based segmentation method in the reconstructed 3D volume is applied, despite some major disadvantages. With this publication, the potential of shifting the segmentation task to a learning-based, view-consistent 2D projection-based method on the downstream MAR's outcome is investigated. For segmenting the present metal, a rather simple learning-based 2D projection-wise segmentation network that is trained using real data acquired during cadaver studies, is examined. To overcome the disadvantages that come along with a 2D projection-wise segmentation, a Consistency Filter is proposed. The influence of the shifted segmentation domain is investigated by comparing the results of the standard fsMAR with a modified fsMAR version using the new segmentation masks. With a quantitative and qualitative evaluation on real cadaver data, the investigated approach showed an increased MAR performance and a high insensitivity against metal artifacts. For cases with metal outside the reconstruction's FoV or cases with vanishing metal, a significant reduction in artifacts could be shown. Thus, increases of up to roughly 3 dB w.r.t. the mean PSNR metric over all slices and up to 9 dB for single slices were achieved. The shown results reveal a beneficial influence of the shift to a 2D-based segmentation method on real data for downstream use with a MAR method, like the fsMAR., Comment: Accepted for publication at the Journal of Machine Learning for Biomedical Imaging (MELBA)

Fiducial markers are commonly used in navigation assisted minimally invasive spine surgery (MISS) and they help transfer image coordinates into real world coordinates. In practice, these markers might be located outside the field-of-view (FOV), due to the limited detector sizes of C-arm cone-beam computed tomography (CBCT) systems used in intraoperative surgeries. As a consequence, reconstructed markers in CBCT volumes suffer from artifacts and have distorted shapes, which sets an obstacle for navigation. In this work, we propose two fiducial marker detection methods: direct detection from distorted markers (direct method) and detection after marker recovery (recovery method). For direct detection from distorted markers in reconstructed volumes, an efficient automatic marker detection method using two neural networks and a conventional circle detection algorithm is proposed. For marker recovery, a task-specific learning strategy is proposed to recover markers from severely truncated data. Afterwards, a conventional marker detection algorithm is applied for position detection. The two methods are evaluated on simulated data and real data, both achieving a marker registration error smaller than 0.2 mm. Our experiments demonstrate that the direct method is capable of detecting distorted markers accurately and the recovery method with task-specific learning has high robustness and generalizability on various data sets.

We explore the sensitivity of directly testing the muon-Higgs coupling at a high-energy muon collider. This is strongly motivated if there exists new physics that is not aligned with the Standard Model Yukawa interactions which are responsible for the fermion mass generation. We illustrate a few such examples for physics beyond the Standard Model. With the accidentally small value of the muon Yukawa coupling and its subtle role in the high-energy production of multiple (vector and Higgs) bosons, we show that it is possible to measure the muon-Higgs coupling to an accuracy of ten percent for a 10 TeV muon collider and a few percent for a 30 TeV machine by utilizing the three boson production, potentially sensitive to a new physics scale about $\Lambda \sim 30-100$ TeV., Comment: Correct the preprint number

We introduce a new definition of speculative bubbles in discrete-time models based on the discounted stock price losing mass at some finite drop-down under an equivalent martingale measure. We provide equivalent probabilistic characterisations of this definition and give examples of discrete-time martingales that are speculative bubbles and those that are not. In the Markovian case, we provide sufficient analytic conditions for the presence of speculative bubbles. We also show that the existence of speculative bubbles is directly linked to the existence of a non-trivial solution to a linear Volterra integral equation of the second kind involving the Markov kernel. Finally, we show that our definition of speculative bubbles in discrete time is consistent with the strict local martingale definition of speculative bubbles in continuous time in the sense that a properly discretised strict local martingale in continuous time is a speculative bubble in discrete time.

This article summarizes the talk given at the LCWS 2021 conference on the status and news of the WHIZARD Monte Carlo event generator. We presented its features relevant for the physics program of future lepton and especially linear colliders as well as recent developments towards including NLO perturbative corrections and a UFO interface to study models beyond the Standard Model. It takes as reference the version 3.0.0$\beta$ released in August 2020 and additionally discusses the developments that will be included in the next major version 3.0.0 to be released in April 2021., Comment: 11 pages, 4 figures, Talk presented at the International Workshop on Future Linear Colliders (LCWS2021), 15-18 March 2021. C21-03-15.1

Metal implants that are inserted into the patient's body during trauma interventions cause heavy artifacts in 3D X-ray acquisitions. Metal Artifact Reduction (MAR) methods, whose first step is always a segmentation of the present metal objects, try to remove these artifacts. Thereby, the segmentation is a crucial task which has strong influence on the MAR's outcome. This study proposes and evaluates a learning-based patch-wise segmentation network and a newly proposed Consistency Check as post-processing step. The combination of the learned segmentation and Consistency Check reaches a high segmentation performance with an average IoU score of 0.924 on the test set. Furthermore, the Consistency Check proves the ability to significantly reduce false positive segmentations whilst simultaneously ensuring consistent segmentations., Comment: Accepted for Bildverarbeitung f\"ur die Medizin, 07.-09.03.2021

If the Standard Model is understood as the first term of an effective field theory, the anomaly-cancellation conditions have to be worked out and fulfilled order by order in the effective field-theory expansion. We bring attention to this issue and study in detail a subset of the anomalies of the effective field theories at the electroweak scale. The end result is a set of sum rules for the operator coefficients. These conditions, which are necessary for the internal consistency of the theory, lead to a number of phenomenological consequences when implemented in analyses of experimental data. In particular, they not only decrease the number of free parameters in different physical processes but have the potential to relate processes with different flavor content. Conversely, a violation of these conditions would necessarily imply the existence of undetected non-decoupling new physics associated with the electroweak energy scale., Comment: 5 pages, 2 figures

We study a microscopic limit order book model, in which the order dynamics depend on the current best bid and ask price and the current volume density functions, simultaneously, and derive its macroscopic high-frequency dynamics. As opposed to the existing literature on scaling limits for limit order book models, we include price changes which do not scale with the tick size in our model to account for large price movement, being for example triggered by highly unforeseen events. Our main result states that, when the size of an individual limit order and the tick size tend to zero while the order arrival rate tends to infinity, the microscopic limit order book model dynamics converge to two one-dimensional jump diffusion processes describing the prices coupled with two infinite dimensional fluid processes describing the standing volumes at the buy and sell side., Comment: 46 pages

Many optoelectronic devices based on organic materials require rapid and long-range singlet exciton transport. Key factors that control the transport of singlet excitons includes the electronic structure of the material, disorder and exciton-phonon coupling. An important parameter whose influence on exciton transport has not been explored is the symmetry of the singlet electronic state (S1). Here, we employ femtosecond transient absorption spectroscopy and microscopy to reveal the relationship between the symmetry of S1 and exciton transport in highly aligned, near-disorder free, one-dimensional conjugated polymers based on polydiacetylene., Comment: 4 figures

Metal artifacts present a frequent challenge to cone-beam CT (CBCT) in image-guided surgery, obscuring visualization of metal instruments and adjacent anatomy. Recent advances in mobile C-arm systems have enabled 3D imaging capacity with non-circular orbits. We extend a previously proposed metal artifacts avoidance (MAA) method to reduce the influence of metal artifacts by prospectively defining a non-circular orbit that avoids metal-induced biases in projection domain. Accurate geometric calibration is an important challenge to accurate 3D image reconstruction for such orbits. We investigate the performance of interpolation-based calibration from a library of circular orbits for any non-circular orbit. We apply the method to non-circular scans acquired for MAA, which involves: (i) coarse 3D localization of metal objects via only two scout views using an end-to-end trained neural network; (ii) calculation of the metal-induced x-ray spectral shift for all possible views; and (iii) identification of the non-circular orbit that minimizes the variations in spectral shift. Non-circular orbits with interpolation-based geometric calibration yielded reasonably accurate 3D image reconstruction. The end-to-end neural network accurately localized metal implants with just two scout views even in complex anatomical scenes, improving Dice coefficient by ~42% compared to a more conventional cascade of separately trained U-nets. In a spine phantom with pedicle screw instrumentation, non-circular orbits identified by the MAA method reduced the magnitude of metal "blomming" artifacts (apparent width of the screw shaft) in CBCT reconstructions by ~70%. The proposed imaging and calibration methods present a practical means to improve image quality in mobile C-arm CBCT by identifying non-circular scan protocols that improve sampling and reduce metal-induced biases in the projection data., Comment: This work was presented at the 6th International Conference on Image Formation in X-Ray Computed Tomography, August, 2020, Regensburg, Germany

A cyclic ordering of the points in a Mendelsohn triple system of order $v$ (or MTS$(v)$) is called a sequencing. A sequencing $D$ is $\ell$-good if there does not exist a triple $(x,y,z)$ in the MTS$(v)$ such that (1) the three points $x,y,$ and $z$ occur (cyclically) in that order in $D$; and (2) $\{x,y,z\}$ is a subset of $\ell$ cyclically consecutive points of $D$. In this paper, we prove some upper bounds on $\ell$ for MTS$(v)$ having $\ell$-good sequencings and we prove that any MTS$(v)$ with $v \geq 7$ has a $3$-good sequencing. We also determine the optimal sequencings of every MTS$(v)$ with $v \leq 10$.

A Mendelsohn triple system of order $v$ (or MTS$(v)$) is a decomposition of the complete graph into directed 3-cyles. We denote the directed 3-cycle with edges $(x,y)$, $(y,z)$ and $(z,x)$ by $(x,y,z)$, $(y,z,x)$ or $(z,x,y)$. An $\ell$-good sequencing of a MTS$(v)$ is a permutation of the points of the design, say $[x_1 \; \cdots \; x_v]$, such that, for every triple $(x,y,z)$ in the design, it is not the case that $x = x_i$, $y = x_j$ and $z = x_k$ with $i < j < k$ and $k-i+1 \leq \ell$; or with $j < k < i$ and $i-j+1 \leq \ell$; or with $k < i < j$ and $j-k+1 \leq \ell$., Comment: 121 pages

A directed triple system of order $v$ (or, DTS$(v)$) is decomposition of the complete directed graph $\vec{K_v}$ into transitive triples. A $v$-good sequencing of a DTS$(v)$ is a permutation of the points of the design, say $[x_1 \; \cdots \; x_v]$, such that, for every triple $(x,y,z)$ in the design, it is not the case that $x = x_i$, $y = x_j$ and $z = x_k$ with $i < j < k$. We prove that there exists a DTS$(v)$ having a $v$-good sequencing for all positive integers $v \equiv 0,1 \bmod {3}$. Further, for all positive integers $v \equiv 0,1 \bmod {3}$, $v \geq 7$, we prove that there is a DTS$(v)$ that does not have a $v$-good sequencing. We also derive some computational results concerning $v$-good sequencings of all the nonisomorphic DTS$(v)$ for $v \leq 7$.

A directed triple system of order $v$ (or, DTS$(v)$) is a decomposition of the complete directed graph $\vec{K_v}$ into transitive triples. An $\ell$-good sequencing of a DTS$(v)$ is a permutation of the points of the design, say $[x_1 \; \cdots \; x_v]$, such that, for every triple $(x,y,z)$ in the design, it is $not$ the case that $x = x_i$, $y = x_j$ and $z = x_k$ with $i < j < k$ and $k-i+1 \leq \ell$. In this report we provide a maximum $\ell$-good sequencing for each DTS$(v)$, $v \leq 7$., Comment: 305 pages

A partial Steiner triple system of order n is sequenceable if there is a sequence of length n of its distinct points such that no proper segment of the sequence is a union of point-disjoint blocks. We prove that if a partial Steiner triple system has at most three point-disjoint blocks, then it is sequenceable., Comment: 18 pages

A partial Steiner triple system is is $sequenceable$ if the points can be sequenced so that no proper segment can be partitioned into blocks. We show that, if $0 \leq a \leq (n-1)/3$, then there exists a nonsequenceable PSTS$(n)$ of size $\frac{1}{3}\binom{n}{2}-a$, for all $n \equiv 1 \pmod{6}$ except for $n=7$.

Given an STS(v), we ask if there is a permutation of the points of the design such that no $\ell$ consecutive points in this permutation contain a block of the design. Results are obtained in the cases $\ell = 3,4$.