Construction of rational surfaces yielding good codes
Copyright policy )Construction of rational surfaces yielding good codes
In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound "à la Weil" of Aubry and Perret together with the bound of Homma and Kim for plane curves. The parameters of several codes from rational surfaces are computed. Among them, the codes defined by the evaluation of forms of degree 3 on an elliptic quadric are studied. As far as we know, such codes have never been treated before. Two other rational surfaces are studied and very good codes are found on them. In particular, a [57,12,34] code over $\mathbf{F}_7$ and a [91,18,53] code over $\mathbf{F}_9$ are discovered, these codes beat the best known codes up to now.
20 pages, 7 figures
FOS: Computer and information sciences, MSC: 94B27, Rational surfaces, Computer Science - Information Theory, plane curves, Algebraic coding theory; cryptography (number-theoretic aspects), Linear systems, Applications to coding theory and cryptography of arithmetic geometry, Theoretical Computer Science, Plane curves, [MATH.MATH-IT] Mathematics [math]/Information Theory [math.IT], Mathematics - Algebraic Geometry, Algebraic Geometry codes, Rational and ruled surfaces, Varieties over finite and local fields, FOS: Mathematics, Rational points, Number Theory (math.NT), 14J26, Algebraic Geometry (math.AG), Engineering(all), Algebra and Number Theory, Mathematics - Number Theory, Applied Mathematics, Information Theory (cs.IT), [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], linear systems, [MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT], rational surfaces, algebraic geometry codes, 14C20, 94B27, 14J26, 11G25, 14C20, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], 004, rational points, [INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT], Finite fields, 11G25, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], [INFO.INFO-IT] Computer Science [cs]/Information Theory [cs.IT], finite fields, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT], Geometric methods (including applications of algebraic geometry) applied to coding theory
FOS: Computer and information sciences, MSC: 94B27, Rational surfaces, Computer Science - Information Theory, plane curves, Algebraic coding theory; cryptography (number-theoretic aspects), Linear systems, Applications to coding theory and cryptography of arithmetic geometry, Theoretical Computer Science, Plane curves, [MATH.MATH-IT] Mathematics [math]/Information Theory [math.IT], Mathematics - Algebraic Geometry, Algebraic Geometry codes, Rational and ruled surfaces, Varieties over finite and local fields, FOS: Mathematics, Rational points, Number Theory (math.NT), 14J26, Algebraic Geometry (math.AG), Engineering(all), Algebra and Number Theory, Mathematics - Number Theory, Applied Mathematics, Information Theory (cs.IT), [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], linear systems, [MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT], rational surfaces, algebraic geometry codes, 14C20, 94B27, 14J26, 11G25, 14C20, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], 004, rational points, [INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT], Finite fields, 11G25, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], [INFO.INFO-IT] Computer Science [cs]/Information Theory [cs.IT], finite fields, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT], Geometric methods (including applications of algebraic geometry) applied to coding theory
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