A new variant of the Compressed Sensing problem is investigated when the number of measurements c... more A new variant of the Compressed Sensing problem is investigated when the number of measurements corrupted by errors is upper bounded by some value l but there are no more restrictions on errors. We prove that in this case it is enough to make 2(t + l) measurements, where t is the sparsity of original data. Moreover for this case a rather simple recovery algorithm is proposed. An analog of the Singleton bound from coding theory is derived what proves optimality of the corresponding measurement matrices.
We present here a combinatorial approach to special divisors and secant divisors on curves over f... more We present here a combinatorial approach to special divisors and secant divisors on curves over finite fields based on their relations with error-correcting codes. Applying to linear systems on curves over finite fields numerous coding theory bounds one gets a lot of bounds on their dimensions. This approach works for curves with many rational points, which leads, for example, to certain ameliorations of Clifford's theorem for such curves.
We show that the lattice Hadwiger (=kissing) number of superballs is exponential in the dimension... more We show that the lattice Hadwiger (=kissing) number of superballs is exponential in the dimension. The same is true for some more general convex bodies.
Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword wit... more Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword without having to access each of the remaining coordinates. A subfamily of LRC codes with hierarchical locality (H-LRC codes) provides added flexibility to the construction by introducing several tiers of recoverability for correcting different numbers of erasures. We present a general construction of codes with 2-level hierarchical locality from maps between algebraic curves and specialize it to several code families obtained from quotients of curves by a subgroup of the automorphism group, including rational, elliptic, Kummer, and Artin-Schreier curves. We further address the question of H-LRC codes with availability, and suggest a general construction of such codes from fiber products of curves. Detailed calculations of parameters for H-LRC codes with availability are performed for Reed-Solomon-and Hermitian-like code families. Finally, we construct asymptotically good families of H-LRC codes from curves related to the Garcia-Stichtenoth tower.
Moscow Journal of Combinatorics and Number Theory, 2019
We construct a sequence of lattices {L ni ⊂ R ni } for n i −→ ∞, with exponentially large kissing... more We construct a sequence of lattices {L ni ⊂ R ni } for n i −→ ∞, with exponentially large kissing numbers, namely, log 2 τ (L ni) > 0.0338 • n i − o(n i). We also show that the maximum lattice kissing number τ l n in n dimensions verifies log 2 τ l n > 0.0219 • n − o(n).
2015 IEEE International Symposium on Information Theory (ISIT), 2015
A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the enc... more A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) other symbols. A family of linear LRC codes that generalize the classic construction of Reed-Solomon codes was constructed in a recent paper by I.
ABSTRACT In this paper we prove two facts on the spectrum of algebraic-geometric (AG-)codes. Afte... more ABSTRACT In this paper we prove two facts on the spectrum of algebraic-geometric (AG-)codes. After recalling some basic definitions and results in Section 2 we estimate the deviation of the spectrum of an AG-code from the spectrum of a random code. In Section 3 it is shown that the spectrum of a general (or “random”) code on a curve with asymptotically maximal number of rational points can be effectively estimated (from above). Moreover in many cases these estimates are better than the spectrum of a random code. In the paper [G. L. Katsman, M. A. Tsfasman and S. G. Vladut, Lect. Notes Math. 1518, 82-98 (1992; Zbl 0772.94014)] one finds an application of these results to the error probability of decoding.
Chapter 6. Cubic Minimal Cones 6.1. Brief overview of the main results 6.2. Algebraic minimal hyp... more Chapter 6. Cubic Minimal Cones 6.1. Brief overview of the main results 6.2. Algebraic minimal hypercones in R n and radial eigencubics 6.3. Metrized and Freudenthal-Springer algebras 6.4. Radial eigencubic algebras iii iv CONTENTS 6.5. Polar and Clifford REC algebras 6.6. The harmonicity of radial eigencubics 6.7. The Peirce decomposition of a REC algebra 6.8. Triality systems and a hidden Clifford algebra structure 6.9. Jordan algebra V • c 6.10. Reducible REC algebras 6.11. Exceptional REC algebras Chapter 7. Singular Solutions in Calibrated Geometries 7.1. Calibrated geometries 7.2. Singular coassociative 4-folds 7.3. Singular solutions of special Lagrangian equations 7.4. More singular solutions and a failure of the maximum principle
We show that for any ε ∈]0, 1[ there exists an analytic outside zero solution to a uniformly elli... more We show that for any ε ∈]0, 1[ there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B ⊂ R 5 which belongs to C 1,ε (B) \ C 1,ε+ (B).
Proceedings. IEEE International Symposium on Information Theory
This talk is devoted to some applications of algebraic geometry to coding theory other than algeb... more This talk is devoted to some applications of algebraic geometry to coding theory other than algebraic geometry codes. The general scheme of these applications is converse to the Goppa construction, which associates a code to some algebraic geometry data. Here, on the contrary, some problems in coding theory give rise to certain algebraic varieties over finite fields, so that these problems can be formulated as questions about these varieties (usually concerning their ratonal points). One schould mention that usually the algebraic geometry problems arising in this way are rather subtle; nevertheless there are some cases where it is possible to solve them using powerful technique of modern algebraic geometry whih leads to rather interesting results in coding theory. In this talk we consider the following results:
According to a celebrated conjecture of Gauss, there are infinitely many real quadratic fields wh... more According to a celebrated conjecture of Gauss, there are infinitely many real quadratic fields whose ring of integers is principal. We recall this conjecture in the framework of global fields. If one removes any assumption on the degree, this leads to various related problems for which we give solutions; namely, we prove that there are infinite families of principal rings of algebraic functions in positive characteristic, which are extensions of a given one, and with prescribed Galois, or ramification, properties, at least in some particular cases.
put forward the following conjecture: Let {C n } be a sequence of binary linear codes of distance... more put forward the following conjecture: Let {C n } be a sequence of binary linear codes of distance d n and A dn be the number of vectors of weight d n in C n. Then log 2 A dn =o(n). We disprove this by constructing a family of linear codes from geometric Goppa codes in which the number of vectors of minimum weight grows exponentially with the length.
Journal de Mathématiques Pures et Appliquées, 2008
We prove the existence of a viscosity solution of a fully nonlinear elliptic equation in 24 dimen... more We prove the existence of a viscosity solution of a fully nonlinear elliptic equation in 24 dimensions with blowing up second derivative.
ABSTRACT It is proved that for algebraic-geometric codes on a curve over F q for q ⩾3... more ABSTRACT It is proved that for algebraic-geometric codes on a curve over F q for q ⩾37 or on a curve of sufficiently large genus over Fq for q ⩾16 there exists a polynomial decoding algorithm up to ( d *-1)/2 errors, d * being the designed minimum distance
A new variant of the Compressed Sensing problem is investigated when the number of measurements c... more A new variant of the Compressed Sensing problem is investigated when the number of measurements corrupted by errors is upper bounded by some value l but there are no more restrictions on errors. We prove that in this case it is enough to make 2(t + l) measurements, where t is the sparsity of original data. Moreover for this case a rather simple recovery algorithm is proposed. An analog of the Singleton bound from coding theory is derived what proves optimality of the corresponding measurement matrices.
We present here a combinatorial approach to special divisors and secant divisors on curves over f... more We present here a combinatorial approach to special divisors and secant divisors on curves over finite fields based on their relations with error-correcting codes. Applying to linear systems on curves over finite fields numerous coding theory bounds one gets a lot of bounds on their dimensions. This approach works for curves with many rational points, which leads, for example, to certain ameliorations of Clifford's theorem for such curves.
We show that the lattice Hadwiger (=kissing) number of superballs is exponential in the dimension... more We show that the lattice Hadwiger (=kissing) number of superballs is exponential in the dimension. The same is true for some more general convex bodies.
Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword wit... more Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword without having to access each of the remaining coordinates. A subfamily of LRC codes with hierarchical locality (H-LRC codes) provides added flexibility to the construction by introducing several tiers of recoverability for correcting different numbers of erasures. We present a general construction of codes with 2-level hierarchical locality from maps between algebraic curves and specialize it to several code families obtained from quotients of curves by a subgroup of the automorphism group, including rational, elliptic, Kummer, and Artin-Schreier curves. We further address the question of H-LRC codes with availability, and suggest a general construction of such codes from fiber products of curves. Detailed calculations of parameters for H-LRC codes with availability are performed for Reed-Solomon-and Hermitian-like code families. Finally, we construct asymptotically good families of H-LRC codes from curves related to the Garcia-Stichtenoth tower.
Moscow Journal of Combinatorics and Number Theory, 2019
We construct a sequence of lattices {L ni ⊂ R ni } for n i −→ ∞, with exponentially large kissing... more We construct a sequence of lattices {L ni ⊂ R ni } for n i −→ ∞, with exponentially large kissing numbers, namely, log 2 τ (L ni) > 0.0338 • n i − o(n i). We also show that the maximum lattice kissing number τ l n in n dimensions verifies log 2 τ l n > 0.0219 • n − o(n).
2015 IEEE International Symposium on Information Theory (ISIT), 2015
A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the enc... more A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) other symbols. A family of linear LRC codes that generalize the classic construction of Reed-Solomon codes was constructed in a recent paper by I.
ABSTRACT In this paper we prove two facts on the spectrum of algebraic-geometric (AG-)codes. Afte... more ABSTRACT In this paper we prove two facts on the spectrum of algebraic-geometric (AG-)codes. After recalling some basic definitions and results in Section 2 we estimate the deviation of the spectrum of an AG-code from the spectrum of a random code. In Section 3 it is shown that the spectrum of a general (or “random”) code on a curve with asymptotically maximal number of rational points can be effectively estimated (from above). Moreover in many cases these estimates are better than the spectrum of a random code. In the paper [G. L. Katsman, M. A. Tsfasman and S. G. Vladut, Lect. Notes Math. 1518, 82-98 (1992; Zbl 0772.94014)] one finds an application of these results to the error probability of decoding.
Chapter 6. Cubic Minimal Cones 6.1. Brief overview of the main results 6.2. Algebraic minimal hyp... more Chapter 6. Cubic Minimal Cones 6.1. Brief overview of the main results 6.2. Algebraic minimal hypercones in R n and radial eigencubics 6.3. Metrized and Freudenthal-Springer algebras 6.4. Radial eigencubic algebras iii iv CONTENTS 6.5. Polar and Clifford REC algebras 6.6. The harmonicity of radial eigencubics 6.7. The Peirce decomposition of a REC algebra 6.8. Triality systems and a hidden Clifford algebra structure 6.9. Jordan algebra V • c 6.10. Reducible REC algebras 6.11. Exceptional REC algebras Chapter 7. Singular Solutions in Calibrated Geometries 7.1. Calibrated geometries 7.2. Singular coassociative 4-folds 7.3. Singular solutions of special Lagrangian equations 7.4. More singular solutions and a failure of the maximum principle
We show that for any ε ∈]0, 1[ there exists an analytic outside zero solution to a uniformly elli... more We show that for any ε ∈]0, 1[ there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B ⊂ R 5 which belongs to C 1,ε (B) \ C 1,ε+ (B).
Proceedings. IEEE International Symposium on Information Theory
This talk is devoted to some applications of algebraic geometry to coding theory other than algeb... more This talk is devoted to some applications of algebraic geometry to coding theory other than algebraic geometry codes. The general scheme of these applications is converse to the Goppa construction, which associates a code to some algebraic geometry data. Here, on the contrary, some problems in coding theory give rise to certain algebraic varieties over finite fields, so that these problems can be formulated as questions about these varieties (usually concerning their ratonal points). One schould mention that usually the algebraic geometry problems arising in this way are rather subtle; nevertheless there are some cases where it is possible to solve them using powerful technique of modern algebraic geometry whih leads to rather interesting results in coding theory. In this talk we consider the following results:
According to a celebrated conjecture of Gauss, there are infinitely many real quadratic fields wh... more According to a celebrated conjecture of Gauss, there are infinitely many real quadratic fields whose ring of integers is principal. We recall this conjecture in the framework of global fields. If one removes any assumption on the degree, this leads to various related problems for which we give solutions; namely, we prove that there are infinite families of principal rings of algebraic functions in positive characteristic, which are extensions of a given one, and with prescribed Galois, or ramification, properties, at least in some particular cases.
put forward the following conjecture: Let {C n } be a sequence of binary linear codes of distance... more put forward the following conjecture: Let {C n } be a sequence of binary linear codes of distance d n and A dn be the number of vectors of weight d n in C n. Then log 2 A dn =o(n). We disprove this by constructing a family of linear codes from geometric Goppa codes in which the number of vectors of minimum weight grows exponentially with the length.
Journal de Mathématiques Pures et Appliquées, 2008
We prove the existence of a viscosity solution of a fully nonlinear elliptic equation in 24 dimen... more We prove the existence of a viscosity solution of a fully nonlinear elliptic equation in 24 dimensions with blowing up second derivative.
ABSTRACT It is proved that for algebraic-geometric codes on a curve over F q for q ⩾3... more ABSTRACT It is proved that for algebraic-geometric codes on a curve over F q for q ⩾37 or on a curve of sufficiently large genus over Fq for q ⩾16 there exists a polynomial decoding algorithm up to ( d *-1)/2 errors, d * being the designed minimum distance
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Papers by SERGE VLADUTS