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The focus of this thesis is the study and construction of covering arrays, relying on maximal period sequences and other tools from finite fields. A covering array of strength $t$, denoted $\mathrm{CA}(N; t, k,v)$, is an $N\times k$ array…
Scale-free percolation is a percolation model on $\mathbb{Z}^d$ which can be used to model real-world networks. We prove bounds for the graph distance in the regime where vertices have infinite degrees. We fully characterize transience vs.…
A general construction of binary sequences with low autocorrelation are considered in the paper. Based on recent progresses about this topic and this construction, several classes of binary sequences with optimal autocorrelation and other…
In this paper, a new construction of interference-free zero correlation zone (IF-ZCZ) sequence sets is proposed by well designed finite Zak transform lattice tessellation. Each set is characterized by the period of sequences $KM^2$, the set…
A perfect isometry is an important relation between blocks of finite groups as many information about blocks are preserved by it. If we consider the group of all perfect isometries between a block to itself then this gives another…
Constant dimension codes (CDCs) have become an important object in coding theory due to their application in random network coding. The multilevel construction is one of the most effective ways to construct constant dimension codes. The…
We show that an appropriate generalization of the oriented area function is a perfect Morse function on the space of three-dimensional configurations of an equilateral polygonal linkage with odd number of edges. Therefore cyclic equilateral…
Recently, an intriguing family of the one-point toric conformal blocks AGT related to the $\mathcal{N}=2^*\,\, SU(2)$ Nekrasov functions was discovered by M. Beccaria and G. Macorini. Members of the family are distinguished by having only…
We study perfect multiple coverings in translation invariant graphs with vertex set $\mathbb{Z}^2$ using an algebraic approach. In this approach we consider any such covering as a two-dimensional binary configuration which we then express…
A circulant-based spatially-coupled (SC) code is constructed by partitioning the circulants in the parity-check matrix of a block code into several components and piecing copies of these components in a diagonal structure. By connecting…
In this paper, we study the sequences with (non-consecutive) two zero-symbols and ideal autocorrelation, which are also known as almost $m$-ary nearly perfect sequences. We show that these sequences are equivalent to $\ell$-partial direct…
Array configurations built on a hierarchy of simple elements have excellent properties for cross-correlation imaging interferometers including a smooth distribution of measured Fourier components, high angular resolution, low side lobes,…
Using evolutionary algorithm and first-principles calculations, we predict a family group of two-dimensional node-line semimetals MX (M=Pd, Pt; X=S, Se, Te), which has zig-zag type mono-layer structure in Pmm2 layer group. Band structure…
Augmented designs are typically used in early-stage breeding programs to compare single replicates of test entries by combining them with replicated check varieties. One or two dimensional incomplete blocking can be incorporated in the…
In this paper, we define a new construction of completely scrambled 0-dimensional systems using the inverse limit of sequences of directed graph covers. These examples are transitive and are not locally equicontinuous.
Low-discrepancy sequences have seen widespread adoption in computer graphics thanks to their superior convergence rates. Since rendering integrals often comprise products of lower-dimensional integrals, recent work has focused on developing…
The second neighbor correlation functions of the spin-${{1/2}}$ $XXZ$ chain in the ground state are expressed in the form of three dimensional integrals. We show that these integrals can be reduced to one-dimensional ones and thereby…
We introduce two-dimensional toroidal arrays that are a variant of the de Bruijn tori. We call them nested perfect toroidal arrays. Instead of asking that every array of a given size has exactly one occurrence, we partition the positions in…
Golay complementary pairs (GCPs) and complete complementary codes (CCCs) have found a wide range of practical applications in coding, signal processing and wireless communication due to their ideal correlation properties. In fact, binary…
By using totally isotropic subspaces in an orthogonal space Omega^{+}(2i,2), several infinite families of packings of 2^k-dimensional subspaces of real 2^i-dimensional space are constructed, some of which are shown to be optimal packings. A…