Related papers: Doppler Tolerance, Complementary Code Sets and the…
Pseudo-Boolean constraints, also known as 0-1 Integer Linear Constraints, are used to model many real-world problems. A common approach to solve these constraints is to encode them into a SAT formula. The runtime of the SAT solver on such…
In this paper, we present an iterative construction of a polar code and develop properties of the dual of a polar code. Based on this approach, belief propagation of a polar code can be presented in the context of low-density parity check…
To define a minimal mathematical framework for isolating some of the characteristic properties of quantum entanglement, we introduce a generalization of the tensor product of graphs. Inspired by the notion of a density matrix, the…
In this paper, we prove that the sub-field images of generalized Reed-Solomon (RS) codes can achieve the symmetric capacity of p-ary memoryless channels. Unlike the totally random linear code ensemble, as a class of maximum distance…
First we generalize the Thue-Morse sequence (the generalized Thue-Morse sequences) by a cyclic permutations and p-adic system, and consider the necessary-sufficient condition that it is non-periodic. Moreover if the generalized Thue-Morse…
We present a Morse-theoretic characterization of collapsibility for 2-dimensional acyclic simplicial complexes by means of the values of normalized optimal combinatorial Morse functions.
A relation between the deformed Hulth\'en potential and the Eckart one is used to write the bound-state wavefunctions of the former in terms of Jacobi polynomials and to calculate their normalization coefficients. The shape invariance…
Golay sequences are well suited for the use as codewords in orthogonal frequency-division multiplexing (OFDM), since their peak-to-mean envelope power ratio (PMEPR) in q-ary phase-shift keying (PSK) modulation is at most 2. It is known that…
We prove that the Thue--Morse sequence $\mathbf t$ along subsequences indexed by $\lfloor n^c\rfloor$ is normal, where $1<c<3/2$. That is, for $c$ in this range and for each $\omega\in\{0,1\}^L$, where $L\geq 1$, the set of occurrences of…
This is the first of a series of papers that develop a systematic bridge between constructions in discrete mathematics and the corresponding continuous analogs. In this paper, we establish an equivalence between Forman's discrete Morse…
We prove a generalization of the digital binomial theorem by constructing a one-parameter subgroup of generalized Sierpinski matrices. In addition, we derive new formulas for the coefficients of Prouhet-Thue-Morse polynomials and describe…
In this paper, we summarize a general method of transforming DG structures into higher structures on the various complexes related to the reduced bar resolution of a given quiver algebra using algebraic Morse theory. As an application, we…
The level of distribution of a complex valued sequence $b$ measures "how well $b$ behaves" on arithmetic progressions $nd+a$. Determining whether $\theta$ is a level of distribution for $b$ involves summing a certain error over $d\leq D$,…
The spectral test of random number generators (R.R. Coveyou and R.D. McPherson, 1967) is generalized. The sequence of random numbers is analyzed explicitly not just via their n-tupel distributions. The generalized analysis of many…
This paper introduces a reformulation of the classical convergence theorem for spectral sequences of filtered complexes which provides an algorithm to effectively compute the induced filtration on the total (co)homology, as soon as the…
We present a three-dimensional generalization of a renormalization group decoding algorithm for topological codes with Abelian anyonic excitations that we previously introduced for two dimensions. This 3D implementation extends our previous…
Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive…
In a recent breakthrough, Kelley and Meka (FOCS 2023) obtained a strong upper bound on the density of sets of integers without nontrivial three-term arithmetic progressions. In this work, we extend their result, establishing similar bounds…
In this paper, we develop a structure theory for generalized spectral sequences, which are derived from chain complexes that are filtered over arbitrary partially ordered sets. Also, a more general construction method reminiscent of exact…
We consider the generalized Thue-Morse sequences $(t_n^{(c)})_{n\ge 0}$ ($c \in [0,1)$ being a parameter) defined by $t_n^{(c)} = e^{2\pi i c s_2(n)}$, where $s_2(n)$ is the sum of digits of the binary expansion of $n$. For the polynomials…