A coding theoretic approach to extending designs

AbstractWe introduce the study of designs in a coset of a binary code which can be held by vectors of a fixed weight. If C is a binary [2n, n, d] code with n odd and the words of weights n - 1 and n + 1 hold complementary t-designs, then we show that the vectors of weight n in a coset of weight 1 also hold a t-design. We also show how to “extend” these designs. We then consider designs in cosets of type I self-dual codes, in particular in the shadow. If the vectors of a fixed weight in the code hold t-designs then so do the vectors of a fixed weight in the shadow. For [24k - 2, 12k - 1, 2 + 4k] type I codes, these designs extend to designs in the type II parent code

Graphs, designs and codes related to the n-cube

For integers n 1; k 0, and k n, the graph k n has vertices the 2n vectors of Fn 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular 1 n is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k D 1; 2; 3, following the results obtained for the binary codes of the n-cube in Fish [Washiela Fish, Codes from uniform subset graphs and cyclic products, Ph.D. Thesis, University of the Western Cape, 2007] and Key and Seneviratne [J.D. Key, P. Seneviratne, Permutation decoding for binary self-dual codes from the graph Qn where n is even, in: T. Shaska, W. C Huffman, D. Joyner, V. Ustimenko (Eds.), Advances in Coding Theory and Cryptology, in: Series on Coding Theory and Cryptology, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007, pp. 152 159 ]. We find the automorphism groups of the graphs and of their associated neighbourhood designs for k D 1; 2; 3, and the dimensions of the ternary codes for k D 1; 2. We also obtain 3-PD-sets for the self-dual binary codes from 2 n when n 0 .mod 4/, n 8

Graphs, designs and codes related to the n-cube

For integers n 1; k 0, and k n, the graph k n has vertices the 2n vectors of Fn 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular 1 n is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k D 1; 2; 3, following the results obtained for the binary codes of the n-cube in Fish [Washiela Fish, Codes from uniform subset graphs and cyclic products, Ph.D. Thesis, University of the Western Cape, 2007] and Key and Seneviratne [J.D. Key, P. Seneviratne, Permutation decoding for binary self-dual codes from the graph Qn where n is even, in: T. Shaska, W. C Huffman, D. Joyner, V. Ustimenko (Eds.), Advances in Coding Theory and Cryptology, in: Series on Coding Theory and Cryptology, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007, pp. 152 159 ]. We find the automorphism groups of the graphs and of their associated neighbourhood designs for k D 1; 2; 3, and the dimensions of the ternary codes for k D 1; 2. We also obtain 3-PD-sets for the self-dual binary codes from 2 n when n 0 .mod 4/, n 8

Computational Aspects of Retrieving a Representation of an Algebraic Geometry Code

Producción CientíficaCode-based cryptography is an interesting alternative to classic number-theoretic public key cryptosystem since it is conjectured to be secure against quantum computer attacks. Many families of codes have been proposed for these cryptosystems such as algebraic geometry codes. In [Designs, Codes and Cryptography, pages 1-16, 2012] -for so called very strong algebraic geometry codes $\mathcal C=C_L(\mathcal X, \mathcal P, E)$, where $\mathcal X$ is an algebraic curve over $\mathbb F_q$, $\mathcal P$ is an $n$-tuple of mutually distinct $\mathbb F_q$-rational points of $\mathcal X$ and $E$ is a divisor of $\mathcal X$ with disjoint support from $\mathcal P$ --- it was shown that an equivalent representation $\mathcal C=C_L(\mathcal Y, \mathcal Q, F)$ can be found. The $n$-tuple of points is obtained directly from a generator matrix of $\mathcal C$, where the columns are viewed as homogeneous coordinates of these points. The curve $\mathcal Y$ is given by $I_2(\mathcal Y)$, the homogeneous elements of degree $2$ of the vanishing ideal $I(\mathcal Y)$. Furthermore, it was shown that $I_2(\mathcal Y)$ can be computed efficiently as the kernel of certain linear map. What was not shown was how to get the divisor $F$ and how to obtain efficiently an adequate decoding algorithm for the new representation. The main result of this paper is an efficient computational approach to the first problem, that is getting $F$. The security status of the McEliece public key cryptosystem using algebraic geometry codes is still not completely settled and is left as an open problemThis research was partly supported by the Danish National Research Foundation and the National Science Foundation of China (Grant No.\ 11061130539) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography and by Spanish grants MTM2007-64704, MTM2010-21580-C02-02 and MTM2012-36917-C03-03. Part of the research of the second author is also funded by the Vernon Wilson Endowed Chair at Eastern Kentucky University during his sabbatical leave

Low thrust chemical rocket technology

An on-going technology program to improve the performance of low thrust chemical rockets for spacecraft on-board propulsion applications is reviewed. Improved performance and lifetime is sought by the development of new predictive tools to understand the combustion and flow physics, introduction of high temperature materials and improved component designs to optimize performance, and use of higher performance propellants. Improved predictive technology is sought through the comparison of both local and global predictions with experimental data. Predictions are based on both the RPLUS Navier-Stokes code with finite rate kinetics and the JANNAF methodology. Data were obtained with laser-based diagnostics along with global performance measurements. Results indicate that the modeling of the injector and the combustion process needs improvement in these codes and flow visualization with a technique such as 2-D laser induced fluorescence (LIF) would aid in resolving issues of flow symmetry and shear layer combustion processes. High temperature material fabrication processes are under development and small rockets are being designed, fabricated, and tested using these new materials. Rhenium coated with iridium for oxidation protection was produced by the Chemical Vapor Deposition (CVD) process and enabled an 800 K increase in rocket operating temperature. Performance gains with this material in rockets using Earth storable propellants (nitrogen tetroxide and monomethylhydrazine or hydrazine) were obtained through component redesign to eliminate fuel film cooling and its associated combustion inefficiency while managing head end thermal soakback. Material interdiffusion and oxidation characteristics indicated that the requisite lifetimes of tens of hours were available for thruster applications. Rockets were designed, fabricated, and tested with thrusts of 22, 62, 440 and 550 N. Performance improvements of 10 to 20 seconds specific impulse were demonstrated. Higher performance propellants were evaluated: Space storable propellants, including liquid oxygen (LOX) as the oxidizer with nitrogen hydrides or hydrocarbon as fuels. Specifically, a LOX/hydrazine engine was designed, fabricated, and shown to have a 95 pct theoretical c-star which translates into a projected vacuum specific impulse of 345 seconds at an area ratio of 204:1. Further performance improvment can be obtained by the use of LOX/hydrogen propellants, especially for manned spacecraft applications, and specific designs must be developed and advanced through flight qualification

Evaluation codes defined by finite families of plane valuations at infinity

We construct evaluation codes given by weight functions defined over polynomial rings in m a parts per thousand yen 2 indeterminates. These weight functions are determined by sets of m-1 weight functions over polynomial rings in two indeterminates defined by plane valuations at infinity. Well-suited families in totally ordered commutative groups are an important tool in our procedureSupported by Spain Ministry of Education MTM2007-64704 and Bancaixa P1-1B2009-03. The authors thank to the referees for their valuable suggestions.Galindo Pastor, C.; Monserrat Delpalillo, FJ. (2014). Evaluation codes defined by finite families of plane valuations at infinity. Designs, Codes and Cryptography. 70(1-2):189-213. https://doi.org/10.1007/s10623-012-9738-7S189213701-2Abhyankar S.S.: Local uniformization on algebraic surfaces over ground field of characteristic p ≠ 0. Ann. Math. 63, 491–526 (1956)Abhyankar S.S.: On the valuations centered in a local domain. Am. J. Math. 78, 321–348 (1956)Abhyankar S.S.: Lectures on expansion techniques in algebraic geometry. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57. Tata Institute of Fundamental Research, Bombay (1977).Abhyankar S.S.: On the semigroup of a meromorphic curve (part I). In: Proceedings of the International Symposium on Algebraic Geometry (Kyoto) Kinokunio Tokio, pp. 249–414 (1977).Abhyankar S.S., Moh T.T.: Newton-Puiseux expansion and generalized Tschirnhausen transformation (I). J. Reine Angew. Math. 260, 47–83 (1973)Abhyankar S.S., Moh T.T.: Newton-Puiseux expansion and generalized Tschirnhausen transformation (II). J. Reine Angew. Math. 261, 29–54 (1973)Berlekamp E.R.: Algebraic Coding Theory. McGraw-Hill, New York (1968)Campillo A., Farrán J.I.: Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models. Finite Fields Appl. 6, 71–92 (2000)Carvalho C., Munuera C., Silva E., Torres F.: Near orders and codes. IEEE Trans. Inf. Theory 53, 1919–1924 (2007)Decker W., Greuel G.M., Pfister G., Schöenemann H.: Singular 3.1.3, a computer algebra system for polynomial computations (2011) http://www.singular.uni-kl.de .Feng G.L., Rao T.R.N.: Decoding of algebraic geometric codes up to the designed minimum distance. IEEE Trans. Inf. Theory 39, 37–45 (1993)Feng G.L., Rao T.R.N.: A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inf. Theory 40, 1003–1012 (1994)Feng G.L., Rao T.R.N.: Improved geometric Goppa codes, part I: basic theory. IEEE Trans. Inf. Theory 41, 1678–1693 (1995)Fujimoto M., Suzuki M.: Construction of affine plane curves with one place at infinity. Osaka J. Math. 39(4), 1005–1027 (2002)Galindo C.: Plane valuations and their completions. Commun. Algebra 23(6), 2107–2123 (1995)Galindo C., Monserrat F.: δ-sequences and evaluation codes defined by plane valuations at infinity. Proc. Lond. Math. Soc. 98, 714–740 (2009)Galindo C., Monserrat F.: The Abhyankar-Moh theorem for plane valuations at infinity. Preprint 2010. ArXiv:0910.2613v2.Galindo C., Sanchis M.: Evaluation codes and plane valuations. Des. Codes Cryptogr. 41(2), 199–219 (2006)Geil O.: Codes based on an $${\mathbb{F}_q}$$ -algebra. PhD Thesis, Aalborg University, June (2000).Geil O., Matsumoto R.: Generalized Sudan’s list decoding for order domain codes. Lecture Notes in Computer Science, vol. 4851, pp. 50–59 (2007)Geil O., Pellikaan R.: On the structure of order domains. Finite Fields Appl. 8, 369–396 (2002)Goppa V.D.: Codes associated with divisors. Probl. Inf. Transm. 13, 22–26 (1997)Goppa V.D.: Geometry and Codes. Mathematics and Its Applications, vol. 24. Kluwer, Dordrecht (1991).Greco S., Kiyek K.: General elements in complete ideals and valuations centered at a two-dimensional regular local ring. In: Algebra, Arithmetic, and Geometry, with Applications, pp. 381–455. Springer, Berlin (2003).Høholdt T., van Lint J.H., Pellikaan R.: Algebraic geometry codes. In: Handbook of Coding Theory, vol. 1, pp. 871–961. Elsevier, Amsterdam (1998).Jensen C.D.: Fast decoding of codes from algebraic geometry. IEEE Trans. Inf. Theory 40, 223–230 (1994)Justesen J., Larsen K.J., Jensen H.E., Havemose A., Høholdt T.: Construction and decoding of a class of algebraic geometric codes. IEEE Trans. Inf. Theory 35, 811–821 (1989)Justesen J., Larsen K.J., Jensen H.E., Høholdt T.: Fast decoding of codes from algebraic plane curves. IEEE Trans. Inf. Theory 38, 111–119 (1992)Massey J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15, 122–127 (1969)Matsumoto R.: Miura’s generalization of one point AG codes is equivalent to Høholdt, van Lint and Pellikaan’s generalization. IEICE Trans. Fundam. E82-A(10), 2007–2010 (1999)Moghaddam M.: Realization of a certain class of semigroups as value semigroups of valuations. Bull. Iran. Math. Soc. 35, 61–95 (2009)O’Sullivan M.E.: Decoding of codes defined by a single point on a curve. IEEE Trans. Inf. Theory 41, 1709–1719 (1995)O’Sullivan M.E.: New codes for the Belekamp-Massey-Sakata algorithm. Finite Fields Appl. 7, 293–317 (2001)Pinkham H.: Séminaire sur les singularités des surfaces (Demazure-Pinkham-Teissier), Course donné au Centre de Math. de l’Ecole Polytechnique (1977–1978).Sakata S.: Extension of the Berlekamp-Massey algorithm to N dimensions. Inf. Comput. 84, 207–239 (1990)Sakata S., Jensen H.E., Høholdt T.: Generalized Berlekamp-Massey decoding of algebraic geometric codes up to half the Feng-Rao bound. IEEE Trans. Inf. Theory 41, 1762–1768 (1995)Sakata S., Justesen J., Madelung Y., Jensen H.E., Høholdt T.: Fast decoding of algebraic geometric codes up to the designed minimum distance. IEEE Trans. Inf. Theory 41, 1672–1677 (1995)Sathaye A.: On planar curves. Am. J. Math. 99(5), 1105–1135 (1977)Shannon C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).Skorobogatov A.N., Vlădut S.G.: On the decoding of algebraic geometric codes. IEEE Trans. Inf. Theory 36, 1051–1060 (1990)Spivakovsky M.: Valuations in function fields of surfaces. Am. J. Math. 112, 107–156 (1990)Suzuki M.: Affine plane curves with one place at infinity. Ann. Inst. Fourier 49(2), 375–404 (1999)Tsfasman S.G., Vlăduţ T.: Zink, modular curves, Shimura curves and Goppa codes, better than Varshamov–Gilbert bound. Math. Nachr. 109, 21–28 (1982)Vlăduţ S.G., Manin Y.I. Linear codes and modular curves. In: Current problems in mathematics, vol. 25, pp. 209–257. Akad. Nauk SSSR Vseoyuz, Moscow (1984).Zariski O.: The reduction of the singularities of an algebraic surface. Ann. Math. 40, 639–689 (1939)Zariski O.: Local uniformization on algebraic varieties. Ann. Math. 41, 852–896 (1940)Zariski O., Samuel P.(1960) Commutative Algebra, vol. II. Springer, Berlin