Papers by Alexander Davydov
IEEE Transactions on Information Theory, 1991
A recent paper shows that the matched-filter/tappeddelay-line structure is optimum not only for l... more A recent paper shows that the matched-filter/tappeddelay-line structure is optimum not only for linear pulse-modulated signals and linear channel distortion, hut also for nonlinear finitealphabet pulse-modulation and some nonlinear channel distortion. This has important practical applications. Therefore, its connection with other work reported in the literature is brought to light in this note.
arXiv (Cornell University), Oct 23, 2022
We consider the structures of the plane-line and point-line incidence matrices of the projective ... more We consider the structures of the plane-line and point-line incidence matrices of the projective space PG(3, q) connected with orbits of planes, points, and lines under the stabilizer group of the twisted cubic. In the literature, lines are partitioned into classes, each of which is a union of line orbits. In this paper, for all q, even and odd, we determine the incidence matrices connected with a family of orbits of the class named O 6. This class contains lines external to the twisted cubic. The considered family include an essential part of all O 6 orbits, whose complete classification is an open problem.
2019 XVI International Symposium "Problems of Redundancy in Information and Control Systems" (REDUNDANCY), 2019
The length function ℓq(r,R) is the smallest length of a q-ary linear code of covering radius R an... more The length function ℓq(r,R) is the smallest length of a q-ary linear code of covering radius R and codimension (redundancy) r. In this paper, we obtained new upper bounds on ℓq(r, 3), r = 3t+1 ≥ 4, and r = 3t+2 ≥ 5, t ≥ 1. For r = 4, 5 we use the one-to-one correspondence between [n, n − r]qR codes and (R − 1)-saturating sets (e.g. complete arcs) in the projective space PG(r − 1, q). Then, with the help of lift-constructions increasing r, we obtain new upper bounds on ℓq(3t + 1, 3), ℓq(3t + 2, 3). In particular, we show that\begin{equation*}\begin{array}{l}\ell_q (r,3) \lt 2.7\sqrt[3]{\ln q}\cdot q^{(r - 3)/3} ,\,r = 3t + 1 \ge 4,t \ge 1,\,q \le 6553; \\ \ell_q (r,3) \lt 2.9\sqrt[3]{\ln q}\cdot q^{(r - 3)/3} ,\,r = 3t + 2 \ge 5,t \ge 1,\,q \le 839. \end{array}\end{equation*}Also, in PG(3, q) we consider an iterative step-by-step construction of complete arcs and prove that uncovered points are evenly placed on the space. A natural conjecture on an estimate of the number of new covered points in every step is done. Under this conjecture, the following bounds for values of q, not limited from above, are proposed:\begin{equation*}\ell_q (r,3) \lt 3\sqrt[3]{\ln q}\cdot q^{(r - 3)/3} ,\,r = 3t + 1 \ge 4,\,t \ge 1.\end{equation*}
The length function \(\ell _q(r,R)\) is the smallest length of a q-ary linear code of covering ra... more The length function \(\ell _q(r,R)\) is the smallest length of a q-ary linear code of covering radius R and codimension r. New upper bounds on \(\ell _q(r,2)\) are obtained for odd \(r\ge 3\). In particular, using the one-to-one correspondence between linear codes of covering radius 2 and saturating sets in the projective planes over finite fields, we prove that $$\begin{aligned} \ell _q(3,2)\le \sqrt{q(3\ln q+\ln \ln q)}+\sqrt{\frac{q}{3\ln q}}+3 \end{aligned}$$ and then obtain estimations of \(\ell _q(r,2)\) for all odd \(r\ge 5\). The new upper bounds are smaller than the previously known ones. Also, the new bounds hold for all q, not necessary large, whereas the previously best known estimations are proved only for q large enough.
Mediterranean Journal of Mathematics
In the projective space PG(3, q), we consider the orbits of lines under the stabilizer group of t... more In the projective space PG(3, q), we consider the orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of PG(3, q) are partitioned into classes, each of which is a union of line orbits. In this paper, all classes of lines consisting of a unique orbit are found. For the remaining line types, with one exception, it is proved that they consist exactly of two or three orbits; sizes and structures of these orbits are determined. Also, the subgroups of the stabilizer group of the twisted cubic fixing lines of the orbits are obtained. Problems which remain open for one type of lines are formulated and, for 5 ≤ q ≤ 37 and q = 64, a solution is provided.
Let [ ] denote a linear code over with length , codimension , and covering radius . We use a modi... more Let [ ] denote a linear code over with length , codimension , and covering radius . We use a modification of constructions of [2 +1 2 3] 2 and [3 +1 3 5] 3 codes ( 5) to produce infinite families of good codes with covering radius 2 and 3 and codimension .
Problems of Information Transmission, 1989
Cornell University - arXiv, Jul 5, 2020
The weight of a coset of a code is the smallest Hamming weight of any vector in the coset. For a ... more The weight of a coset of a code is the smallest Hamming weight of any vector in the coset. For a linear code of length n, we call integral weight spectrum the overall numbers of weight w vectors, 0 ≤ w ≤ n, in all the cosets of a fixed weight. For maximum distance separable (MDS) codes, we obtained new convenient formulas of integral weight spectra of cosets of weight 1 and 2. Also, we give the spectra for the weight 3 cosets of MDS codes with minimum distance 5 and covering radius 3.
2017 IEEE International Symposium on Information Theory (ISIT), 2017
We consider the weight spectrum of a class of quasiperfect binary linear codes with code distance... more We consider the weight spectrum of a class of quasiperfect binary linear codes with code distance 4. For example, extended Hamming code and Panchenko code are the known members of this class. Also, it is known that in many cases Panchenko code has the minimal number of weight 4 codewords. We give exact recursive formulas for the weight spectrum of quasi-perfect codes and their dual codes. As an example of application of the weight spectrum we derive a lower estimate for the conditional probability of correction of erasure patterns of high weights (equal to or greater than code distance).
Tables of the currently known parameters of symmetric configurations are given. Formulas for para... more Tables of the currently known parameters of symmetric configurations are given. Formulas for parameters of the known infinite families of symmetric configurations are presented as well. The results of the recent paper [13] are used. This work can be viewed as an appendix to [13], in the sense that the tables given here cover a much larger set of parameters.
The following two important problems are considered in the paper: constructing a low density pari... more The following two important problems are considered in the paper: constructing a low density parity check code on a bipartite graph and rapid encoding of this code. For a given constituent code, the first problem solving is reduced to constructing and investigation of pa- rameters of the matrix describing connections of two vertex subsets of a regular bipartite graph (biadjacency matrix). It is convenient to treat the such matrix as a support-matrix of a code word. We propose a number of constructions that essentially extend the region of accessible parameters of the such matrices including these providing graphs without 4-cycles. Biadjacency matrices of regular bipartite graphs without 4-cycles are treated also as the incidence matrices of symmetric combinatorial configurations. This contributes to understanding and solving of the first problem. The second problem solving leads to search of such support-matrix transforma- tions that maximize the encoding speed and allow us to find ...
Problems of Information Transmission, 2010
We consider sequences in which every symbol of an alphabet occurs at most once. We construct fami... more We consider sequences in which every symbol of an alphabet occurs at most once. We construct families of such sequences as nonlinear subcodes of a q-ary [n, k, n − k + 1] q Reed-Solomon code of length n ≤ q consisting of words that have no identical symbols. We introduce the notion of a bunch of words of a linear code. For dimensions k ≤ 3 we obtain constructive lower estimates (tight bounds in a number of cases) on the maximum cardinality of a subcode for various n and q, and construct subsets of words meeting these estimates and bounds. We define codes with words that have no identical symbols, observe their relation to permutation codes, and state an optimization problem for them.
Problems of Information Transmission, 2011
We consider MIMO communication systems with Rayleigh fading. We propose a new coded modulation ba... more We consider MIMO communication systems with Rayleigh fading. We propose a new coded modulation based on orthogonal sequences and state a new decodability condition. We introduce concepts and constructions of permutation free (PF) and permutation and repetition free (PRF) codes. We also propose a construction of PRF codes with sign manipulation, whose code rate can exceed 1. For better analysis and construction of these codes we introduce a one-to-one mapping that transforms signal matrices to vectors over a finite field. We propose construction algorithms for PF and PRF codes. We build PF and PRF codes with large cardinality, which in several case achieve the maximum cardinality. Simulation of the constructed codes and estimation of their performance was done in Simulink environment. Results show high error-correcting capability, which often reaches that of STBC codes with full transmit diversity.
IEEE Transactions on Information Theory, 1991
Finite Fields and Their Applications, 2002
Finite "eld towers GF(q.) are considered, where P"p L p L 2 p LR R and all primes p G are distinc... more Finite "eld towers GF(q.) are considered, where P"p L p L 2 p LR R and all primes p G are distinct factors of (q!1). Under this condition irreducible binomials of the form x.!c can be used for recursive extension of "nite "elds. We give description of an in"nite sequence of irreducible binomials, new e!ective algorithms for fast multiplication and inversion in the tower, and "nite and asymptotic estimates of arithmetic complexity. It is important that the achievable asymptotic estimate of the complexity has the form O (log Q logKlog Q), Q"q., where log 5 51 and is the minimal factor of q!1.
In a projective plane Π _q (not necessarily Desarguesian) of order q, a point subset S is saturat... more In a projective plane Π _q (not necessarily Desarguesian) of order q, a point subset S is saturating (or dense) if any point of Π _q∖ S is collinear with two points in S. Using probabilistic methods, the following upper bound on the smallest size s(2,q) of a saturating set in Π _q is proved: s(2,q)≤ 2√((q+1) (q+1))+2 2√(q q). We also show that for any constant c> 1 a random point set of size k in Π _q with 2c√((q+1)(q+1))+2< k<q^2-1/q+2 q is a saturating set with probability greater than 1-1/(q+1)^2c^2-2. Our probabilistic approach is also applied to multiple saturating sets. A point set S⊂Π_q is (1,μ)-saturating if for every point Q of Π _q∖ S the number of secants of S through Q is at least μ , counted with multiplicity. The multiplicity of a secant ℓ is computed as #(ℓ ∩ S)2. The following upper bound on the smallest size s_μ(2,q) of a (1,μ)-saturating set in Π_q is proved: s_μ(2,q)≤ 2(μ +1)√((q+1) (q+1))+2 2(μ +1)√( q q) for 2≤μ≤√(q). By using inductive constructions, u...
Advances in Mathematics of Communications, 2022
The length function \begin{document}$ \ell_q(r,R) $\end{document} is the smallest length of a \be... more The length function \begin{document}$ \ell_q(r,R) $\end{document} is the smallest length of a \begin{document}$ q $\end{document}-ary linear code with codimension (redundancy) \begin{document}$ r $\end{document} and covering radius \begin{document}$ R $\end{document}. In this work, new upper bounds on \begin{document}$ \ell_q(tR+1,R) $\end{document} are obtained in the following forms: \begin{document}$ \begin{equation*} \begin{split} &(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &\phantom{(a)\; } q\;{\rm{ is \;an\; arbitrary \;prime\; power}},\; c{\rm{ \;is\; independent \;of\; }}q. \end{split} \end{equation*} $\end{document} \begin{document}$ \begin{equation*} \begin{split} &(b)\; \ell_q(r,R)< 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &\phantom{(b)\; } q\;{\rm{ is \;an\; arbitrary\; prime \;power}},\; q\;{\rm{ is \;large\; enough}}. \end{split} \end{equation*} $\end{document} In the literature, for \begin{doc...
Uploads
Papers by Alexander Davydov