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  • Cited by 6
Publisher:
Cambridge University Press
Online publication date:
October 2017
Print publication year:
2017
Online ISBN:
9780511982170

Book description

This well-balanced text touches on theoretical and applied aspects of protecting digital data. The reader is provided with the basic theory and is then shown deeper fascinating detail, including the current state of the art. Readers will soon become familiar with methods of protecting digital data while it is transmitted, as well as while the data is being stored. Both basic and advanced error-correcting codes are introduced together with numerous results on their parameters and properties. The authors explain how to apply these codes to symmetric and public key cryptosystems and secret sharing. Interesting approaches based on polynomial systems solving are applied to cryptography and decoding codes. Computer algebra systems are also used to provide an understanding of how objects introduced in the book are constructed, and how their properties can be examined. This book is designed for Masters-level students studying mathematics, computer science, electrical engineering or physics.

Reviews

'The book under review is intended as an introduction to the field for beginning graduate students. The authors do a good job of covering a wide range of topics and keeping the discussion detailed while still as elementary as one can hope to make it.'

Darren Glass Source: MAA Reviews

'While 'coding' may commonly connote confidential communication and security for sensitive data, coding also enters the engineering of information transmission and retrieval, simply for efficient resilience against mechanical error and corrupting noise. From these two purposes rise the two distinct subjects of cryptology and error-correction, receiving here an unusual, unified treatment. Good codes spring from diverse directions, since so many branches of mathematics inform their development: combinatorics, linear algebra, finite fields, ring theory, algebraic geometry, and computer algebra. The girth of this volume reflects the reasonably detailed exposition of all this background material, most of it likely new to engineering students (but students of pure mathematics should also read this book for practical applications of seemingly abstract material they have likely studied). The authors maintain a high level of rigor, keeping all proofs short by astute organization without ever stinting on detail.'

D. V. Feldman Source: Choice

'This book provides a fine exposition of the topics to those students who are novices to the field. At the same time it will also be of interest to readers who are already familiar with some of the concepts discussed in the book. It provides a valuable schematic summary and consolidated overview of the field.'

S. V. Nagaraj Source: SIGACT News

I was impressed by the scope of the book: many topics in algebraic coding theory are addressed and now collected in one book. Someone reading the entire book, will obtain a very good overview of algebraic coding theory.

Peter Beelen Source: Nieuw Archief voor Weskunde

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Contents

References
[1] Abramson, N. 1963. Information theory and coding. New York: McGraw-Hill.
[2] Agrawal, M., Kayal, N. and Saxena, N. 2004. PRIMES is in P. Ann. ofMath. (2), 160(2), 781–793.
[3] Aigner, M. 1979. Combinatorial theory. New York: Springer.
[4] Albrecht, M. R. and Cid, C. 2008. Algebraic techniques in differentialcryptanalysis. Pages 55–60 of: Proceedings of the First International Conference on Symbolic Computation and Cryptography, Beijing, China.
[5] Albrecht, M. R., Cid, C., Faugere, J.-C. and Perret, L. 2012. On the relation between the MXL family of algorithms and Gröbner basis algorithms. J. Symb. Comput., 47(8), 926–941.
[6] Aleshnikov, I., Deolalikar, V., Kumar, P. V. and Stichtenoth, H. 2001. Towards a basis for the space of regular functions in a tower of function fields meeting the Drinfeld-Vladut bound. Pages 14–24 of: Finite fields and applications (Augsburg, 1999). Berlin: Springer.
[7] Ardila, F. 2007. Computing the Tutte polynomial of a hyperplane arrangement. Pacific J. Math., 230, 1–26.
[8] Arimoto, S. 1962. On a non-binary error-correcting code. Inform. Process. Japan, 2, 22–23.
[9] Ashikhmin, A. and Barg, A. 1998. Minimal vectors in linear codes. IEEE Trans. Inform. Theory, 44(5), 2010–2017.
[10] Assmus, E. F., Mattson, H.F. and Turyn, R. 1967. Cyclic codes. Air Force Cambridge Research Labs, Report AFCRL-67-0365.
[11] Athanasiadis, C. A. 1996. Characteristic polynomials of subspace arrangements and finite fields. Adv. Math., 122, 193–233.
[12] Augot, D., Bardet, M. and Faugère, J.-C. 2009. On the decoding of cyclic codes with Newton identities. J. Symb. Comp., 44(12), 1608–1625.
[13] Ball, S. 2012. On sets of vectors of a finite vector space in which every subset of basis size is a basis. J. Eur. Math. Soc. (JEMS), 14(3), 733–748.
[14] Bansal, N., Pendavingh, R. A. and Pol, J. G. van der. 2015. On the number of matroids. Combinatorica, 35(3), 253–277.
[15] Bard, G. V. 2009. Algebraic cryptanalysis. Dordrecht: Springer.
[16] Barg, A. 1993. At the dawn of the theory of codes. Math. Intell., 15, 2–26.
[17] Barg, A. 1997. The matroid of supports of a linear code. Appl. Algebra Eng. Comm. Comput., 8, 165–172.
[18] Barg, A. 1998. Complexity issues in coding theory. Pages 649–756 of: Handbook of coding theory, vol. 1. Elsevier.
[19] Bassa, A., Beelen, P., Garcia, A. and Stichtenoth, H. 2015. Towers of function fields over non-prime finite fields. Mosc. Math. J., 15(1), 1–29, 181.
[20] Bassalygo, L. A. 1965. New upper bounds for error-correcting codes. Probl. Peredaci Inform., 1(vyp. 4), 41–44.
[21] Becker, A., Joux, A., May, A. and Meurer, A. 2012. Decoding random binary linear codes in 2n/20: how 1+1 = 0 improves information set decoding. Pages 520–536 of: Advances in Cryptology –EUROCRYPT 2012. Lecture Notes in Computer Science, vol. 7237 Heidelberg: Springer.
[22] Beelen, P. and Hoholdt, T. 2008a. The decoding of algebraic geometry codes. Pages 49–98 of: Advances in algebraic geometry codes. Series on Coding Theory and Cryptology, vol. 5 Hackensack, NJ: World Scientific Publishing.
[23] Beelen, P. and Hoholdt, T. 2008b. List decoding using syndromes. Pages 315–331 of: Algebraic geometry and its applications. Series on Number Theory and its Applications, vol. 5 Hackensack, NJ: World Scientific Publishing.
[24] Berlekamp, E. R. 1973. Goppa codes. IEEE Trans. Inform. Theory, IT-19, 590–592.
[25] Berlekamp, E. R. 1974. Key papers in the development of coding theory. New York: IEEE Press.
[26] Berlekamp, E. R. 1984. Algebraic coding theory. Laguna Hills, CA: Aegon Park Press.
[27] Berlekamp, E. R., McEliece, R. J. and van Tilborg, H. C. A. 1978. On the inherent intractability of certain coding problems. IEEE Trans. Inform. Theory, 24, 384–386.
[28] Bernstein, D. J., Lange, T. and Peters, C. 2008. Attacking and defending the McEliece cryptosystem. Pages 31–46 of: Post-Quantum Cryptography: Second International Workshop, PQCrypto 2008, proceedings. Lecture Notes in Computer Science, vol. 5299 Berlin: Springer.
[29] Bernstein, D. J, Buchmann, J. and Dahmen, E. 2009. Post-quantum cryptography. Berlin: Springer.
[30] Bernstein, D. J., Lange, T. and Peters, C. 2011. Smaller decoding exponents: ball-collision decoding. Pages 743–760 of: Advances in Cryptology –CRYPTO 2011. Lecture Notes in Computer Science, vol. 6841. Springer, Heidelberg.
[31] Bierbrauer, J., Johansson, T., Kabatianskii, G. and Smeets, B. 1994. On families of hash functions via geometric codes and concatenation. Pages 331–342 of: Advances in Cryptology –CRYPTO 93. Lecture Notes in Computer Science, vol. 773 Berlin: Springer.
[32] Biggs, N. 1993. Algebraic graph theory. Cambridge University Press.
[33] Biham, E. and Shamir, A. 1990. Differential cryptanalysis of DES-like cryptosystems. Pages 2–21 of: Advances in cryptology –CRYPTO 90. Lecture Notes in Computer Science, vol. 537 Berlin: Springer.
[34] Birkhoff, G. 1930. On the number of ways of coloring a map. Proc. Edinburgh Math. Soc., 2, 83–91.
[35] Birkhoff, G. 1935. Abstract linear dependence and lattices. Amer. J. Math., 56, 800–804.
[36] Björner, A. and Ekedahl, T. 1997. Subarrangments over finite fields: Chomological and enumerative aspects. Adv. Math., 129, 159–187.
[37] Blackburn, J. E., Crapo, H. and Higgs, D. A. 1973. A catalogue of combinatorial geometries. Math. Comput., 27, 155–166.
[38] Blahut, R. E. 1983. Theory and practice of error control codes. Reading: Addison-Wesley.
[39] Blahut, R. E. 2003. Algebraic codes for data transmission. Cambridge University Press.
[40] Blahut, R. E. 2008. Algebraic codes on lines, planes, and curves: an engineering approach. Cambridge University Press.
[41] Blake, I. F. 1973. Algebraic coding theory: History and development. Stroudsburg: Dowden, Hutchinson and Ross.
[42] Blakely, G. R. 1979. Safeguarding cryptographic keys. Pages 313–317 of: Proceedings of 1979 National Computer Conference.
[43] Blass, A., and Sagan, B.E. 1997. Möbius functions of lattices. Adv. Math., 129, 94–123.
[44] Blum, L., Shub, M. and Smale, S. 1989. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. (N.S.), 21(1), 1–46.
[45] Boer, M. A. de. 1996. Almost MDS Codes. Des. Codes Cryptography, 9(2), 143–155.
[46] Boer, M. A. de and Pellikaan, R. 1999. Gröbner bases for codes. Chap. 10, pages 237–259 of: Some tapas of computer algebra. Berlin: Springer- Verlag.
[47] Bogdanov, A., Knudsen, L. R., Leander, G., Paar, C., Poschmann, A., Robshaw, M. J., Seurin, Y. and Vikkelsoe, C. 2007. PRESENT: an ultra-lightweight block cipher. Pages 450–466 of: Cryptographic Hardware and Embedded Systems –CHES 2007, 9th International Workshop, proceedings. Lecture Notes in Computer Science, no. 4727. Springer.
[48] Bogdanov, A., Khovratovich, D. and Rechberger, C. 2011. Biclique cryptanalysis of the full AES. Pages 344–371 of: Advances in Cryptology –ASIACRYPT 2011 –17th International Conference on the Theory and Application of Cryptology and Information Security, proceedings. 7073. Berlin: Springer.
[49] Boppana, R. B. and Sipser, M. 1990. The complexity of finite functions. Pages 757–804 of: Handbook of theoretical computer science, vol. Amsterdam: Elsevier.
[50] Borges-Quintana, M., Borges-Trenard, M. A., Fitzpatrick, P. and Martínez-Moro, E. 2008. Gröbner bases and combinatorics for binary codes. Appl. Algebra Eng. Comm. Comput., 19(5), 393–411.
[51] Bose, R. C. and Bush, K. A. 1952. Orthogonal arrays of strength two and three. Ann. Math. Statistics, 23, 508–524.
[52] Bose, R. C., and Ray-Chaudhuri, D. K. 1960. On a class of error correcting binary group codes. Inform. Control, 3, 68–79.
[53] Brickenstein, M. and Bulygin, S. 2008. Attacking AES via solving systems in the key variables only. Pages 118–123 of: Proceedings of the First International Conference on Symbolic Computation and Cryptography, Beijing, China.
[54] Britz, T. 2002. MacWilliams identities and matroid polynomials. The Electronic J. Combin., 9, R19.
[55] Britz, T. 2007. Higher support matroids. Discrete Math., 307, 2300–2308.
[56] Britz, T. and Rutherford, C. G. 2005. Covering radii are not matroid invariants. Discrete Math. 296, 117–120.
[57] Britz, T. and Shiromoto, K. 2008. A MacWilliams type identity for matroids. Discrete Math., 308, 4551–4559.
[58] Brouwer, A. E. 1998. Bounds on the size of linear codes. Pages 295–461 of: Handbook of coding theory, vol. 1. Elsevier.
[59] Bruen, A. A., Thas, J. A. and Blokhuis, A. 1988. On M.D.S. codes, arcs in PG(n, q) with q even, and a solution of three fundamental problems of B. Segre. Invent. Math., 92(3), 441–459.
[60] Brylawski, T. 1972. A decomposition for combinatorial geometries. Trans. Amer. Math. Soc., 171, 235–282.
[61] Brylawski, T. and Oxley, J. 1979. Intersection theory for embeddings of matroids into uniform geometries. Stud. Appl. Math., 61, 211–244.
[62] Brylawski, T. and Oxley, J. 1980. Several identities for the characteristic polynomial of a combinatorial geometry. Discrete Math., 31(2), 161–170.
[63] Brylawski, T. and Oxley, J. 1992. The Tutte polynomial and its applications. Pages 173–226 of: Matroid applications. Cambridge University Press.
[64] Buchberger, B. 1965. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Ph.D. thesis, Universität Innsbruck.
[65] Buchmann, J. 2004. Introduction to cryptography. Berlin: Springer.
[66] Buchmann, J., Pyshkin, A. and Weinmann, R.-P. 2006. A zerodimensional Groebner basis for AES-128. Pages 78–88 of: Fast Software Encryption, 13th International Workshop, FSE 2006, revised selected papers. Lecture Notes in Computer Science, vol. 4047 Berlin: Springer.
[67] Buhler, J. P., Lenstra, H. W. Jr., and Pomerance, C. 1993. Factoring integers with the number field sieve. Pages 50–94 of: The development of the number field sieve. Lecture Notes in Computer Science, vol. 1554 Berlin: Springer.
[68] Bulygin, S. 2009a. Computer algebra in coding theory and cryptanalysis: Polynomial system solving for decoding linear codes and algebraic cryptanalysis. Saarbrücken, Deutschland: Südwestdeutscher Verlag für Hochschulschriften.
[69] Bulygin, S. 2009b. Polynomial system solving for decoding linear codes and algebraic cryptanysis. Ph.D. thesis, Universität Kaiserslautern.
[70] Bulygin, S. and Pellikaan, R. 2009. Bounded distance decoding of linear error-correcting codes with Gröbner bases. J. Symbolic Comp., 44, 1626–1643.
[71] Bulygin, S. and Pellikaan, R. 2010. Decoding and finding the minimum distance with Gröbner bases: history and new insights. Pages 585–622 of: Selected topics in information and coding theory. Series on Coding Theory and Cryptology, vol. 7 Hackensack, NJ: World Scientific Publishing.
[72] Bürgisser, P., Clausen, M. and Shokrollahi, M. A. 1997. Algebraic complexity theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 315 Berlin: Springer-Verlag. With the collaboration of Thomas Lickteig.
[73] Bush, K. A. 1952. Orthogonal arrays of index unity. Ann. Math. Statistics, 23, 426–434.
[74] Caboara, M. and Mora, T. 2002. The Chen-Reed-Helleseth-Truong decoding algorithm and the Gianni-Kalkbrenner Gröbner shape theorem. Appl. Algebra Eng. Comm. Computing, 13(3), 209–232.
[75] Cameron, P. J. and Lint, J. H. van. 1991. Designs, graphs, codes and their links. London Mathematical Society Student Texts, vol. 22. Cambridge University Press.
[76] Carlet, C. 2010. Boolean functions for cryptography and error correcting codes. Pages 257–397 of: Boolean models and methods in mathematics, computer science, and engineering. Cambridge University Press.
[77] Carlitz, L. 1932. The arithmetic of polynomials in a Galois field. Amer. J. Math., 54, 39–50.
[78] Cartier, P. 1981. Les arrangements d'hyperplans: un chapitre de geometrie combinatoire. Seminaire N. Bourbaki, 561, 1–22.
[79] Charpin, P. 1998. Open problems on cyclic codes. Pages 963–1063 of: Handbook of coding theory. Amsterdam: North-Holland.
[80] Chen, H. and Cramer, R. 2006. Algebraic geometric secret sharing schemes and secure multi-party computations over small fields. Pages 521–536 of: Advances in Cryptology –CRYPTO 2006. Lecture Notes in Computer Science, vol. 4117 Berlin: Springer.
[81] Cid, C. and Leurent, G. 2005. An Analysis of the XSL Algorithm. Pages 333–352 of: Advances in Cryptology –ASIACRYPT 2005, 11th International Conference on the Theory and Application of Cryptology and Information Security, proceedings. Lecture Notes in Computer Science, vol. 3788 Berlin: Springer.
[82] Cid, C., Murphy, S. and Robshaw, M. J. B. 2005. Small scale variants of the AES. Pages 145–162 of: Fast Software Encryption: 12th International Workshop, FSE 2005, revised selected papers. Lecture Notes in Computer Science, vol. 3557 Berlin: Springer.
[83] Cid, C., Murphy, S. and Robshaw, M. J. B. 2006. Algebraic aspects of the Advanced Encryption Standard. Springer-Verlag.
[84] Coffey, J. T. and Goodman, R. M. 1990. Any code of which we cannot think is good. IEEE Trans. Inform. Theory, 36(6), 1453–1461.
[85] Cohen, H. and Frey, G. et al. 2012. Handbook of elliptic and hyperelliptic curve cryptography. Second edn. Boca Raton, FL: Chapman & Hall/CRC.
[86] Cook, S. A. 1971. The complexity of theorem proving procedures. Pages 151–158 of: Proceedings of the Third Annual ACM Symposium on Theory of Computing.
[87] Cooper, A. B. 1993. Toward a new method of decoding algebraic codes using Gröbner bases. Pages 1–11 of: Transactions of the Tenth Army Conference on Applied Mathematics and Computing.
[88] Courtois, N. and Pieprzyk, J. 2002. Cryptanalysis of block ciphers with overdefined systems of equations. Pages 267–287 of: Advances in Cryptology –ASIACRYPT 2002, 8th International Conference on the Theory and Application of Cryptology and Information Security, proceedings. Lecture Notes in Computer Science, vol. 2501 Berlin: Springer.
[89] Courtois, N., Klimov, A., Patarin, J. and Shamir, A. 2000. Efficient algorithms for solving overdefined systems of multivariate polynomial equations. Pages 392–407 of: Advances in Cryptology –EUROCRYPT 2000, International Conference on the Theory and Application of Cryptographic Techniques, proceedings. Lecture Notes in Computer Science, vol. 1807 Berlin: Springer.
[90] Cox, D. A., Little, J. and O'Shea, D. 2005. Using algebraic geometry. Second edn. Graduate Texts in Mathematics, vol. 185 New York: Springer.
[91] Cox, D. A., Little, J. and O'Shea, D. 2007. Ideals, varieties, and algorithms. Third edn. Springer-Verlag.
[92] Crapo, H. 1968. Möbius inversion in lattices. Arch. Math., 19, 595–607.
[93] Crapo, H. 1969. The Tutte polynomial. Aequationes Math., 3, 211–229.
[94] Crapo, H. and Rota, G.-C. 1970. On the foundations of combinatorial theory: combinatorial geometries. Cambridge MA: MIT Press.
[95] Daemen, J. and Vincent, R. 2001. The wide trail design strategy. Pages 222–238 of: Cryptography and Coding, 8th IMA International Conference, proceedings. Lecture Notes in Computer Science, vol. 2260 Berlin: Springer.
[96] Daemen, J. and Vincent, R. 2002. The design of Rijndael: AES –The Advanced Encryption Standard. Berlin: Springer.
[97] Delsarte, P. 1973. An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl.
[98] Delsarte, P. 1975. On subfield subcodes of modified Reed-Solomon codes. IEEE Trans. Inform. Theory, IT-21(5), 575–576.
[99] Diffie, W. 1992. The first ten years of public key cryptography. Pages 135–176 of: Contemporary cryptology: The science of information integrity. New York: IEEE Press.
[100] Diffie, W. and Hellman, M. E. 1976. New directions in cryptography. IEEE Trans. Inform. Theory, 22, 644–654.
[101] Dodunekova, R., Dodunekov, S. M. and Klove, T. 1997. Almost-MDS and near-MDS codes for error detection. IEEE Trans. Inform. Theory, 43(1), 285–290.
[102] Dornstetter, J. L. 1987. On the equivalence of the Berlekamp-Massey and the Euclidean algorithms. IEEE Trans. Inform. Theory, 33, 428–431.
[103] Dür, A. 1987. The automorphism groups of Reed-Solomon codes. J. Combin. Theory Ser. A, 44(1), 69–82.
[104] Duursma, I. M. 1993a. Algebraic decoding using special divisors. IEEE Trans. Inform. Theory, 39, 694–698.
[105] Duursma, I. M. 1993b. Decoding codes from curves and cyclic codes. Ph.D. thesis, Eindhoven University of Technology.
[106] Duursma, I. M. 1993c. Majority coset decoding. IEEE Trans. Inform. Theory, 39, 1067–1071.
[107] Duursma, I. M. 1999. Weight distributions of geometric Goppa codes. Trans. Amer. Math. Soc., 351, 3609–3639.
[108] Duursma, I. M. 2001. From weight enumerators to zeta functions. Discrete Appl. Math., 111, 55–73.
[109] Duursma, I. M. 2003. Combinatorics of the two-variable zeta function. Pages 109–136 of: International Conference on Finite Fields and Applications.
[110] Duursma, I. M. 2008. Algebraic geometry codes: general theory. Pages 1–48 of: Advances in algebraic geometry codes. New Jersey: World Scientific.
[111] Duursma, I. M. and Kötter, R. 1994. Error-locating pairs for cyclic codes. IEEE Trans. Inform. Theory, 40, 1108–1121.
[112] Duursma, I. M. and Mak, K.-H. 2013. On lower bounds for the Ihara constants A(2) and A(3). Compos. Math., 149(7), 1108–1128.
[113] Duursma, I. M., and Pellikaan, R. 2006. A symmetric Roos bound for linear codes. J. Combin. Theory Ser. A, 113(8), 1677–1688.
[114] Duursma, I. M, Kirov, R. and Park, S. 2011. Distance bounds for algebraic geometric codes. J. Pure Appl. Algebra, 215(8), 1863–1878.
[115] Ehrhard, D. 1993. Achieving the designed error capacity in decoding algebraic-geometric codes. IEEE Trans. Inform. Theory, 39(3), 743–751.
[116] El Gamal, T. 1985. A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inform. Theory, 31, 469–472.
[117] Elias, P. 1957. List decoding for noisy channels. Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Mass., Rep. No. 335.
[118] Elias, P. 1991. Error-correcting codes for list decoding. IEEE Trans. Inform. Theory, 37(1), 5–12.
[119] Euler, L. 1736. Solutio problematis ad geometriam situs pertinentis. Commentarii Academiae Scientiarum Imperialis Petropolitanae, 8, 128–140.
[120] Farr, J. and Gao, S. 2005. Gröbner bases and generalized Pade approximation. Math. Comput., 75, 461–473.
[121] Faugère, J.-C. 1999. A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra, 139, 61–88.
[122] Faugère, J.-C. 2002. A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). Pages 75–83 of: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation. New York: ACM.
[123] Faugère, J.-C., Otmani, A., Perret, L. and Tillich, J.-P. 2010. Algebraic cryptanalysis of McEliece variants with compact keys. Pages 279–298 of: Advances in Cryptology –EUROCRYPT 2010, 29th Annual International Conference on the Theory and Applications of Cryptographic Techniques, proceedings. Lecture Notes in Computer Science, vol. 6110 Berlin: Springer.
[124] Feng, G.-L. and Rao, T. R. N. 1993a. A class of algebraic geometric codes from curves in high-dimensional projective spaces. Pages 132–146 of: Applied algebra, algebraic algorithms and error-correcting codes (San Juan, PR, 1993). Lecture Notes in Comput. Science, vol. 673 Berlin: Springer.
[125] Feng, G.-L. and Rao, T. R. N. 1994. A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inform. Theory, 40(4), 1003–1012.
[126] Feng, G.-L. and Rao, T. R. N. 1995. Improved geometric Goppa codes. I. Basic theory. IEEE Trans. Inform. Theory, 41(6, part 1), 1678–1693. Special issue on algebraic geometry codes.
[127] Feng, G.-L.,Wei, V. K., Rao, T. R. N. and Tzeng, K. K. 1994. Simplified understanding and efficient decoding of a class of algebraic-geometric codes. IEEE Trans. Inform. Theory, 40(4), 981–1002.
[128] Feng, G. L. and Rao, T. R. N. 1993b. Decoding algebraic-geometric codes up to the designed minimum distance. IEEE Trans. Inform. Theory, 39(1), 37–45.
[129] Fitzgerald, J. 1996. On algebraic decoding of algebraic-geometric and cyclic codes. Ph.D. thesis, Linköping University of Technology.
[130] Fitzgerald, J. and Lax, R. F. 1998. Decoding affine variety codes using Gröbner bases. Design. Code. Cryptogr., 13, 147–158.
[131] Forney, G. D. Jr. 1965. On decoding BCH codes. IEEE Trans. Inform. Theory, IT-11, 549–557.
[132] Forney, G. D. Jr. 1966a. Concatenated codes. Cambridge, MA: The MIT Press. MIT Research Monograph, No. 37.
[133] Forney, G. D. Jr. 1966b. Generalized minimum distance decoding. IEEE Trans. Inform. Theory, IT-12, 125–131.
[134] Fulton, W. 1989. Algebraic curves. Advanced Book Classics. An introduction to algebraic geometry, Notes written with the collaboration of Richard Weiss, Reprint of 1969 original. Redwood City, CA: Addison-Wesley.
[135] García, A. and Stichtenoth, H. 1995a. Algebraic function fields over finite fields with many rational places. IEEE Trans. Inform. Theory, 41(6, part 1), 1548–1563. Special issue on algebraic geometry codes.
[136] García, A. and Stichtenoth, H. 1995b. A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut, bound. Invent. Math., 121(1), 211–222.
[137] García, A. and Stichtenoth, H. 1996. On the asymptotic behaviour of some towers of function fields over finite fields. J. Number Theory, 61(2), 248–273.
[138] Geelen, J., Gerards, B. and Whittle, G. 2013. The highly connected matroids in minor-closed classes. http://arxiv.org/abs/1312.5012.
[139] Geil, O. and Pellikaan, R. 2002. On the structure of order domains. Finite Fields Appl., 8(3), 369–396.
[140] Geil, O., Matsumoto, R. and Ruano, D. 2013. Feng-Rao decoding of primary codes. Finite Fields Appl., 23, 35–52.
[141] Gentry, C. 2009. Fully homomorphic encryption using ideal lattices. Pages 169–178 of: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009.
[142] Gilbert, E. N. 1952. A comparison of signalling alphabets. Bell Syst. Techn. J., 31, 504–522.
[143] Gilbert, E. N., MacWilliams, F. J., and Sloan, N. J. A. 1974. Codes, which detect deception. Bell Syst. Tech. J., 33(3), 405–424.
[144] Golay, M. 1962. Notes on digital coding. Proc. IEEE, 37, 637.
[145] Goppa, V. D. 1970. A new class of linear correcting codes. Probl. Peredaci Inform., 6(3), 24–30.
[146] Goppa, V. D. 1977. Codes associated with divisors. Probl. Inform. Transmission, 13, 22–26.
[147] Goppa, V. D. 1981. Codes on algebraic curves. Soviet Math. Dokl., 24, 170–172.
[148] Goppa, V. D. 1983. Algebraico-geometric codes. Math. USSR Izvestija, 21, 75–91.
[149] Goppa, V. D. 1984. Codes and information. Russian Math. Surveys, 39, 87–141.
[150] Goppa, V. D. 1989. Geometry and codes, mathematics and its applications. Dordrecht: Soviet series 24, Kluwer Academic Publishing.
[151] Gorenstein, D. and Zierler, N. 1961. A class of error-correcting codes in pm symbols. J. Soc. Indust. Appl. Math., 9, 207–214.
[152] Granville, A. 2005. It is easy to determine whether a given integer is prime. Bull. Amer. Math. Soc. (N.S.), 42(1), 3–38.
[153] Greene, C. 1976. Weight enumeration and the geometry of linear codes. Stud. Appl. Math., 55, 119–128.
[154] Greene, C. and Zaslavsky, T. 1983. On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions and orientations of graphs. Trans. Amer. Math. Soc., 280, 97–126.
[155] Greuel, G.-M. and Pfister, G. 2008. A singular introduction to commutative algebra. Second edn. Springer.
[156] Griesmer, J. H. 1960. A bound for error-correcting codes. IBM J. Res. Develop., 4, 532–542.
[157] Guruswami, V. 2001. List decoding of error-correcting codes. Pro- Quest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Massachusetts Institute of Technology.
[158] Guruswami, V. and Sudan, M. 1999. Improved decoding of Reed- Solomon and algebraic-geometry codes. IEEE Trans. Inform. Theory, 45(6), 1757–1767.
[159] Guruswami, V. and Vardy, A. 2005. Maximum-likelihood decoding of Reed-Solomon codes is NP-hard. IEEE Trans. Inform. Theory, 51, 2249–2256.
[160] Hämäläinen, H., Honkala, I., Litsyn, S. and Ostergard, P. R. J. 1995. Football pools –a game for mathematicians. Amer. Math. Monthly, 102, 579–588.
[161] Hamming, R. W. 1950. Error detecting and error correcting codes. Bell Syst. Tech. J., 29, 147–160.
[162] Hamming, R. W. 1980. Coding and information theory. New Jersey: Prentice-Hall.
[163] Hansen, J. P. 1987. Codes on the Klein quartic, ideals, and decoding. IEEE Trans. Inform. Theory, 33(6), 923–925.
[164] Hartmann, C. R. P. and Tzeng, K. K. 1972. Generalizations of the BCH bound. Inform. Contr., 20, 489–498.
[165] Heijnen, P. and Pellikaan, R. 1998. Generalized Hamming weights of q-ary Reed-Muller codes. IEEE Trans. Inform. Theory, 44(1), 181–196.
[166] Helgert, H. J. 1972. Srivastava codes. IEEE Trans. Inform. Theory, IT-18, 292–297.
[167] Helgert, H.J. 1974. Alternant codes. Inform. Contr., 26(4), 369–380.
[168] Helleseth, T., Klove, T. and Mykkeltveit, J. 1977. The weight distribution of irreducible cyclic codes with block lengths n1((ql − 1)/N). Discrete Math., 18, 179–211.
[169] Henocq, T. and Rotillon, D. 1996. The theta divisor of a Jacobian variety and the decoding of geometric Goppa codes. J. Pure Appl. Algebra, 112(1), 13–28.
[170] Hermelina, M. and Nyberg, K. 2000. Correlation properties of the Bluetooth combiner generator. Pages 17–29 of: Information Security and Cryptology, ICISC 1999, Proceedings. Lecture Notes in Computer Science, vol. 1787 Berlin: Springer.
[171] Heytmann, A. E. and Jensen, J. M. 2000. On the equivalence of the Berlekamp-Massey and the Euclidean algorithm for decoding. IEEE Trans. Inform. Theory, 46, 2614–2624.
[172] Hirschfeld, J. W. P. and Storme, L. 1998. The packing problem in statistics, coding theory and finite projective spaces. J. Statist. Plann. Inference, 72(1-2), 355–380. R. C. Bose Memorial Conference (Fort Collins, CO, 1995).
[173] Hirschfeld, J. W. P. and Thas, J. A. 2016. General Galois geometries. Springer Monographs in Mathematics. London: Springer.
[174] Hirschfeld, J. W. P., Korchmaros, G. and Torres, F. 2008. Algebraic curves over a finite field. Princeton Series in Applied Mathematics. Princeton University Press.
[175] Hocquenghem, A. 1959. Codes correcteurs d'erreurs. Chiffres, 2, 147–156.
[176] Hoholdt, T. and Pellikaan, R. 1995. On decoding algebraic-geometric codes. IEEE Trans. Inform. Theory, 41, 1589–1614.
[177] Hoholdt, T., Lint, J. H. van and Pellikaan, R. 1998. Algebraic geometry codes. Pages 871–961 of: Handbook of coding theory, vol. 1 Amsterdam: North-Holland.
[178] Huffman, W. C. 1998. Codes and groups. Pages 1345–1440 of: Handbook of coding theory. Amsterdam: North-Holland.
[179] Huffman, W. C. and Pless, V.S. 1998. Handbook of coding theory. New York: Elsevier.
[180] Huffman, W. C. and Pless, V. S. 2003. Fundamentals of error-correcting codes. Cambridge University Press.
[181] Ihara, Y. 1981. Some remarks on the number of rational points of algebraic curves over finite fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28(3), 721–724.
[182] Johnson, S. M. 1962. A new upper bound for error-correcting codes. IRE Trans. Inform. Theory, 8, 203–207.
[183] Joyner, D., Ksir, A. and Traves, W. 2007. Automorphism groups of generalized Reed-Solomon codes. Pages 114–125 of: Advances in coding theory and cryptography. Series on Coding Theory and Cryptology, vol. 3 Hackensack, NJ: World Scientific Publishing.
[184] Jurrius, R. P. M. J. 2012. Codes, arrangements, matroids, and their polynomial links. Ph.D. thesis, Technical University Eindhoven.
[185] Jurrius, R. P. M. J. 2008. Classifying polynomials of linear codes. M.Sc. thesis, Leiden University.
[186] Jurrius, R. P. M. J. and Pellikaan, R. 2011. Codes, arrangements and matroids. Pages 219–325 of: Algebraic geometry modelling in information theory, Series on Coding Theory and Cryptology, vol. 8 Hackensack, NJ: World Scientific Publishing.
[187] Justesen, J. 1976. On the complexity of decoding Reed-Solomon codes. IEEE Trans. Inform. Theory, 22, 237–238.
[188] Justesen, J. and Hoholdt, T. 2004. A course in error-correcting codes. Zürich: EMS Textbooks in Math.
[189] Justesen, J., Larsen, K. J., Jensen, H. E., Havemose, A. and Hoholdt, T. 1989. Construction and decoding of a class of algebraic geometry codes. IEEE Trans. Inform. Theory, 35(4), 811–821.
[190] Karnin, E. D., Greene, J. W. and Hellman, M.E. 1983. On secret sharing systems. IEEE Trans. Inform. Theory, 29(1), 35–31.
[191] Kasami, T., Lin, S. and Peterson, W. W. 1968. New generalizations of the Reed-Muller codes. I. Primitive codes. IEEE Trans. Inform. Theory, IT-14, 189–199.
[192] Kashyap, N. 2008. A decomposition theory for binary linear codes. IEEE Trans. Inform. Theory, 54(7), 3035–3038.
[193] Katsman, G. L., Tsfasman, M. A. and Vlădut,, S. G. 1984. Modular curves and codes with a polynomial construction. IEEE Trans. Inform. Theory, 30(2, part 2), 353–355.
[194] Katsman, G. L. and Tsfasman, M. A. 1987. Spectra of algebraicgeometric codes. Probl. Peredachi Inform., 23, 19–34.
[195] Katsman, G.L. and Tsfasman, M.A. 1989. A remark on algebraic geometric codes. Pages 197–199 of: Representation theory, group rings, and coding theory. Contemporary Mathematics, vol. 93 Providence, RI: American Mathematical Society.
[196] Katz, J. 2010. Digital signatures. Springer.
[197] Kirfel, C. and Pellikaan, R. 1995. The minimum distance of codes in an array coming from telescopic semigroups. IEEE Trans. Inform. Theory, 41(6, part 1), 1720–1732. Special issue on algebraic geometry codes.
[198] Klein, F. 1878. Ueber die Transformation siebenter Ordnung der elliptischen Functionen. Math. Ann., 14(3), 428–471.
[199] Kleinjung, T., Aoki, K., Franke, J., Lenstra, A. K., Thome, E., Bos, J. W., Gaudry, P., Kruppa, A., Montgomery, P. L., Osvik, D. A., Riele, H. te, Timofeev, A. and Zimmermann, P. 2010. Factorization of a 768-bit RSA Modulus. Pages 333–350 of: Advances in Cryptology –CRYPTO 2010, 30th Annual Cryptology Conference, proceedings. Lecture Notes in Computer Science, no. 6223. Berlin: Springer.
[200] Klove, T. 1978. The weight distribution of linear codes over GF(ql) having generator matrix over GF(q). Discrete Math., 23, 159–168.
[201] Klove, T. 1992. Support weight distribution of linear codes. Discrete Math., 106/107, 311–316.
[202] Knudsen, L. R. and Robshaw, M. 2011. The block cipher companion. Information security and cryptography. Heidelberg, London: Springer.
[203] Kolluru, M. S., Feng, G.-L. and Rao, T. R. N. 2000. Construction of improved geometric Goppa codes from Klein curves and Klein-like curves. Appl. Algebra Engrg. Comm. Comput., 10(6), 433–464.
[204] Kötter, R. 1992. A unified description of an error locating procedure for linear codes. Pages 113–117 of: Proceedings of Algebraic and Combinatorial Coding Theory.
[205] Kung, J. P. S. 1986. A source book in matroid theory. Boston: Birkhäuser.
[206] Lachaud, G. 1986. Les codes geometriques de Goppa. Asterisque, 189–207. Seminar Bourbaki, Vol. 1984/85.
[207] Leonard, D. A. 2001. Finding the defining functions for one-point algebraic-geometry codes. IEEE Trans. Inform. Theory, 47(6), 2566–2573.
[208] Levin, L. A. 1973. Universal search problems. Probl. Peredachi Inform., 9, 115–116.
[209] Lidl, R. and Niederreiter, H. 1994. Introduction to finite fields and their applications. Cambridge University Press.
[210] Lin, S. and Costello, D. J. 1983. Error control coding: fundamentals and applications. New Jersey: Prentice-Hall.
[211] Lint, J. H. van. 1975. A survey on perfect codes. Rocky Mountain J. Math., 5, 215–228.
[212] Lint, J. H. van. 1990. Algebraic geometric codes. Pages 137–162 of: Coding theory and design theory, Part I. IMA Volume in Mathematics and its Applications, vol. 20 New York: Springer.
[213] Lint, J. H. van. 1999. Introduction to coding theory. Graduate Texts in Mathematics, vol. 86. 3rd ed. New York: Springer-Verlag.
[214] Lint, J. H. van and Geer, G. van der. 1988. Introduction to coding theory and algebraic geometry. DMV Seminar, vol. 12 Basel: Birkhäuser Verlag.
[215] Lint, J. H. van and Springer, T. A. 1987. Generalized Reed-Solomon codes from algebraic geometry. IEEE Trans. Inform. Theory, 33(3), 305–309.
[216] Lint, J. H. van and Wilson, R. M. 1986. On the minimum distance of cyclic codes. IEEE Trans. Inform. Theory, 32(1), 23–40.
[217] Lint, J. H. van and Wilson, R. M. 1992. A course in combinatorics. Cambridge University Press.
[218] Loeliger, H.-A. 1994. On the basic averaging arguments for linear codes. Pages 251–261 of: Communications and cryptography: Two sides of one tapestry. Kluwer.
[219] Loeliger, H.-A. 1997. Averaging bounds for lattices and linear codes. IEEE Trans. Inform. Theory, 43(6), 1767–1773.
[220] Lopez, B. 1996. Plane models of Drinfeld modular curves. Ph.D. thesis, Universidad Complutense, Madrid.
[221] Lopez, B. 1999. A special integral basis for a plane model of the Drinfeld modular curve X1(n) mod T. Manuscripta Math., 99(1), 55–72.
[222] MacKay, D. 2003. Information theory, inference and learning algorithms. Cambridge University Press.
[223] MacWilliams, F. J. 1963. A theorem on the distribution of weights in a systematic code. Bell Syst. Tech. J., 42, 79–94.
[224] MacWilliams, F. J. 1968. A historical survey. Pages 3–13 of: Error correcting codes. New York: Wiley.
[225] MacWilliams, F. J. and Sloane, N. J. A. 1977. The theory of errorcorrecting codes. Amsterdam: North-Holland Mathematical Library.
[226] Manin, Yu. 1981. What is the maximum number of points on a curve over F2? J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28(3), 715–720.
[227] Marcolla, C., Orsini, E. and Sala, M. 2012. Improved decoding of affinevariety codes. J. Pure Appl. Algebra, 216(7), 1533–1565.
[228] Marquez-Corbella, I. and Pellikaan, R. 2016. A characterization of MDS codes that have an error correcting pair. Finite Fields Appl., 40, 224–245.
[229] Martínez-Moro, E., Munuera, C. and Ruano, D. 2008. Advances in algebraic geometry codes. Series on Coding Theory and Cryptology, vol. 5 Hackensack, NJ: World Scientific Publishing.
[230] Massey, J. L. 1969. Shift-register synthesis and BCH decoding. IEEE Trans. Inform. Theory, 15, 122–127.
[231] Massey, J. L. 1993. Minimal codewords and secret sharing. Pages 276–279 of: Sixth Joint Swedish-Russian Workshop on Information theory, proceedings.
[232] Massey, J. L. 1995. On some applications of coding theory. Pages 33–47 of: Cryptography, Codes and Ciphers: Cryptography and Coding IV.
[233] Massey, J. L. and Schaub, T. Linear complexity in coding theory. Pages 19–32 of: Coding theory and applications (Cachan, 1986). Lecture Notes in Computer Science, vol. 311 Berlin: Springer.
[234] Matsui, M. 1994. Linear cryptanalysis method for DES cipher. Pages 386–397 of: Advances in Cryptology –EUROCRYPT 1993, proceedings. Lecture Notes in Computer Science, vol. 765 Berlin: Springer.
[235] Mattson, H. F. and Solomon, G. 1961. A new treatment of Bose- Chaudhuri codes. J. Soc. Indust. Appl. Math., 9, 654–669.
[236] McCurley, K. S. 1988. A key distribution system equivalent to factoring. J. Cryptology, 1, 95–105.
[237] McEliece, R. J. 1977. The theory of information and coding. Reading: Addison-Wesley.
[238] McEliece, R. J. 1978. A public-key cryptosystem based on algebraic coding theory. DSN Progress Report, 42–44, 114–116.
[239] McEliece, R. J. and Sawate, D. V. 1981. On sharing secrets and Reed- Solomon codes. Comm. ACM, 24, 583–584.
[240] McEliece, R. J. and Swanson, L. 1994. Reed-Solomon codes and the exploration of the solar system. Pages 25–40 of: Reed-Solomon codes and their applications. New York: IEEE Press.
[241] McEliece, R. J. Rodemich, E. R., Rumsey, H. Jr. and Welch, L. R. 1977. New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities. IEEE Trans. Inform. Theory, IT-23(2), 157–166.
[242] Menezes, A., Oorschot, P. van and Vanstone, S. 1996. Handbook of applied cryptography. Boca Raton, FL: CRC Press.
[243] Miura, S. 1992. Algebraic geometric codes on certain plane curves. IEICE Trans., 75A(11), 1739–1745.
[244] Miura, S. and Kamiya, N. 1993. Geometric-Goppa codes on some maximal curves and their minimum distance. Proc. IEEE Inform. Theory Workshop, 85–86.
[245] Mohamed, M. S. E., Cabarcas, D., Ding, J., Buchmann, J. A. and Bulygin, S. 2009. MXL3: An efficient algorithm for computing Gröbner bases of zero-dimensional ideals. Pages 87–100 of: Information, Security and Cryptology –ICISC 2009, 12th International Conference.
[246] Mohamed, M. S. E., Bulygin, S., Zohner, M., Heuser, A., Walter, M. and Buchmann, J. A. 2013. Improved algebraic side-channel attack on AES. J. Cryptogr. Eng., 3(3), 139–156.
[247] Moreno, C. 1991. Algebraic curves over finite fields. Cambridge Tracts in Mathematics, vol. 97. Cambridge University Press.
[248] Mphako, E. G. 2000. Tutte polynomials of perfect matroid designs. Comb., Probab. Comput., 9, 363–367.
[249] Muller, D.E. 1954. Application of Boolean algebra to switching circuit design and to error detection. IRE Trans. Electron. Comput., 3, 6–12.
[250] Murphy, S. and Robshaw, M. 2002. Essential algebraic structure within the AES. Pages 1–16 of: Advances in Cryptology –CRYPTO 2002, 22nd Annual International Cryptology Conference, proceedings. Lecture Notes in Computer Science, vol. 2442 Berlin: Springer.
[251] Nechvatal, J. 1992. Public key cryptography. Pages 177–288 of: Contemporary cryptology: the science of information integrity. New York: IEEE Press.
[252] Nelson, P. and Zwam, S. H. M. van. 2015. On the existence of asymptotically good linear codes in minor-closed classes. IEEE Trans. Inform. Theory, 61(3), 1153–1158.
[253] Niederreiter, H. and Xing, C. 1998. Towers of global function fields with asymptotically many rational places and an improvement on the Gilbert-Varshamov bound. Math. Nachr., 195, 171–186.
[254] Niederreiter, H. and Xing, C. 2001. Rational points on curves over finite fields: theory and applications. London Mathematical Society Lecture Note Series, vol. 285. Cambridge University Press.
[255] NIST. 1977. Federal Information Standards Publication, Data Encryption Standard (DES).
[256] NIST. 2001. Federal Information Standards Publication, Advanced Encryption Standard (AES).
[257] Orlik, P. and Terao, H. 1992. Arrangements of hyperplanes. Grundlehren der Mathematischen Wissenschaften, vol. 300 Berlin: Springer-Verlag.
[258] Orsini, E. and Sala, M. 2005. Correcting errors and erasures via the syndrome variety. J. Pure Appl. Algebra, 200, 191–226.
[259] Oxley, J. G. 2011. Matroid theory. Second edn. Oxford University Press.
[260] Pellikaan, R. 1988. On decoding linear codes by error correcting pairs. Preprint Eindhoven University of Technology.
[261] Pellikaan, R. 1989. On a decoding algorithm of codes on maximal curves. IEEE Trans. Inform. Theory, 35, 1228–1232.
[262] Pellikaan, R. 1992. On decoding by error location and dependent sets of error positions. Discrete Math., 106–107, 369–381.
[263] Pellikaan, R. 1996a. On the existence of error-correcting pairs. J. Stat. Plann. Infer., 229–242.
[264] Pellikaan, R. 1996b. The shift bound for cyclic, Reed-Muller and geometric Goppa codes. Pages 155–174 of: Arithmetic, geometry and coding theory (Luminy, 1993). Berlin: de Gruyter.
[265] Pellikaan, R. 2001. On the existence of order functions. J. Stat. Plan. Infer., 94, 287–301.
[266] Pellikaan, R. and Wu, X.-W. 2004. List decoding of q-ary Reed-Muller codes. IEEE Trans. Inform. Theory, 50(4), 679–682.
[267] Pellikaan, R., Shen, B.-Z. and Wee, G. J. M. van. 1991. Which linear codes are algebraic-geometric ? IEEE Trans. Inform. Theory, 37, 583–602.
[268] Pellikaan, R., Perret, M. and Vlădut,, S. G. 1996. Arithmetic, geometry and coding theory (Luminy, 1993). Berlin: de Gruyter.
[269] Pellikaan, R., Stichtenoth, H. and Torres, F. 1998. Weierstrass semigroups in an asymptotically good tower of function fields. Finite Fields Appl., 4(4), 381–392.
[270] Peters, C. 2011. Curves, codes, and cryptography. Ph.D. thesis, Eindhoven University of Technology.
[271] Peterson, W. W. 1960. Encoding and error-correction procedures for the Bose-Chaudhuri codes. Trans. IRE, IT-6, 459–470.
[272] Peterson, W. W. and Weldon, E. J. 1972. Error-correcting codes. Cambridge, MA: MIT Press.
[273] Pieprzyk, J. and Zhang, X. M. 2003. Ideal threshold schemes from MDS codes. Pages 269–279 of: Information Security and Cryptology, ICISC 2002, Proceedings. Lecture Notes in Computer Science, vol. 2587 Berlin: Springer.
[274] Pless, V. 1968. On the uniqueness of the Golay codes. J. Comb. Theory, 5, 215–228.
[275] Pless, V. 1982. Introduction to the theory of error-correcting codes. New York: John Wiley & Sons.
[276] Pless, V. 1998. Coding constructions. Pages 141–176 of: Handbook of coding theory, vol. 1 Amsterdam: North-Holland.
[277] Plotkin, M. 1960. Binary codes with specified minimum distance. IRE Trans., IT-6, 445–450.
[278] Pomerance, C. 1990. Factoring. Pages 27–47 of: Cryptology and computational number theory, vol. 42 Rhode Island: American Mathematical Society.
[279] Prange, E. 1962. The use of information sets in decoding cyclic codes. IRE Trans., IT-8, 5–9.
[280] Rabin, M. 1979. Digitalized signatures and public-key functions as intractable as factorization. Tech. rept. MIT/LCS/TR-212. Massachusetts Institute of Technology.
[281] Raddum, H. 2007. MRHS Equation Systems. Pages 232–245 of: Selected Areas in Cryptography, 14th International Workshop, SAC 2007, revised selected papers. Lecture Notes in Computer Science, vol. 4876 Berlin: Springer.
[282] Rao, R. C. 1947. Factorial experiments derivable from combinatorial arrangements of arrays. Suppl. J. Roy. Statist. Soc., 9, 128–139.
[283] Reed, I. S. and Solomon, G. 1960. Polynomial codes over certain finite fields. J. Soc. Indust. Appl. Math., 8, 300–304.
[284] Reid, M. 1988. Undergraduate algebraic geometry. LondonMathematical Society Student Texts, vol. 12. Cambridge University Press.
[285] Retter, C. T. 1976. Bounds on Goppa codes. IEEE Trans. Inform. Theory, 22(4), 476–482.
[286] Rivest, R. L., Shamir, A. and Adleman, L.M. 1977. A method for obtaining digital signatures and public-key cryptosystems. Communications of ACM, 21, 120–126.
[287] Roos, C. 1982. A generalization of the BCH bound for cyclic codes, including the Hartmann-Tzeng bound. J. Combin. Theory Ser. A, 33(2), 229–232.
[288] Roos, C. 1983. A new lower bound for the minimum distance of a cyclic code. IEEE Trans. Inform. Theory, 29(3), 330–332.
[289] Rota, G.-C. 1964. On the foundations of combinatorial theory I: Theory of Möbius functions. Zeitschriften für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2, 340–368.
[290] Roth, R. M. and Ruckenstein, G. 2000. Efficient decoding of Reed- Solomon codes beyond half the minimum distance. IEEE Trans. Inform. Theory, 46(1), 246–257.
[291] Roth, R. M. and Seroussi, G. 1985. On generator matrices of MDS codes. IEEE Trans. Inform. Theory, 31(6), 826–830.
[292] Safavi-Naini, R., Wang, H. and Xing, C. 2001. Linear Authentication Codes: Bounds and Constructions. Pages 127–135 of: Advances in Cryptology –INDOCRYPT 2001, proceedings. Lecture Notes in Computer Science, vol. 2247 Berlin: Springer.
[293] Sarwate, D. 1977. On the complexity of decoding Goppa codes. IEEE Trans. Inform. Theory, 23, 515–516.
[294] Schoof, R. 1992. Algebraic curves over F2 with many rational points. J. Number Theory, 41(1), 6–14.
[295] Schouhamer Immink, K. A. 1994. Reed-Solomon codes and the compact disc. Pages 41–59 of: Reed-Solomon codes and their applications. New York: IEEE Press.
[296] Serre, J.-P. 1983. Sur le nombre des points rationnels d'une courbe algebrique sur un corps fini. C. R. Acad. Sci. Paris Ser. I Math., 296(9), 397–402.
[297] Shamir, A. 1979. How to share a secret. Comm. ACM, 22, 612–613.
[298] Shannon, A. 1948. A mathematical theory of communication. Bell Sys. Tech. J., 27, 379–423 and 623–656.
[299] Shor, P. W. 1997. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput., 26, 1484–1509.
[300] Shoup, V. 1994. Fast construction of irreducible polynomials over finite fields. J. Symbolic Comput., 17(5), 371–391.
[301] Shparlinski, I. E. 1993. Finding irreducible and primitive polynomials. Appl. Algebra Engrg. Comm. Comput., 4(4), 263–268.
[302] Shum, K. W., Aleshnikov, I., Kumar, P. V., Stichtenoth, H. and Deolaikar, V. 2001. A low-complexity algorithm for the construction of algebraic-geometric codes better than the Gilbert-Varshamov bound. IEEE Trans. Inform. Theory, 47(6), 2225–2241.
[303] Simonis, J. 1993. The effective length of subcodes. Appl. Algebra Eng. Comm. Comput., 5, 371–377.
[304] Singleton, R. C. 1964. Maximum distance q-nary codes. IEEE Trans. Inform. Theory, IT-10, 116–118.
[305] Skorobogatov, A. N. 1991. The parameters of subcodes of algebraicgeometric codes over prime subfields. Discrete Appl. Math., 33(1-3), 205–214. Applied algebra, algebraic algorithms, and error-correcting codes (Toulouse, 1989).
[306] Skorobogatov, A. N. 1992. Linear codes, strata of Grassmannians, and the problems of Segre. Pages 210–223 of: Coding theory and algebraic geometry. Lecture Notes in Mathematics, vol. 1518 Berlin: Springer- Verlag.
[307] Slepian, D. 1974. Key papers in the development of information theory. New York: IEEE Press.
[308] Smid, M. E. and Branstad, D. K. 1992. The Data Encryption Standard: Past and Future. Pages 43–64 of: Contemporary cryptology: the science of information integrity. New York: IEEE Press.
[309] Stanley, R. P. 1997. Enumerative combinatorics. Vol. 1. Cambridge University Press.
[310] Stanley, R. P. 2007. An introduction to hyperplane arrangements. Pages 389–496 of: Geometric combinatorics. IAS/Park City Mathematical Series, vol. 13 Providence, RI: American Mathematical Society.
[311] Stepanov, S. A. 1999. Codes on algebraic curves. New York: Kluwer Academic/Plenum Publishers.
[312] Stichtenoth, H. 1988. A note on Hermitian codes over GF(q2). IEEE Trans. Inform. Theory, 34(5, part 2), 1345–1348. Coding techniques and coding theory.
[313] Stichtenoth, H. 1990. On the dimension of subfield subcodes. IEEE Trans. Inform. Theory, 36(1), 90–93.
[314] Stichtenoth, H. 1993. Algebraic function fields and codes. Berlin: Springer.
[315] Stichtenoth, H. and Tsfasman, M. A. 1992. Coding theory and algebraic geometry (Luminy, 1991). Lecture Notes in Mathematics, vol. 1518 Berlin: Springer.
[316] Stinson, D. R. 1990. The combinatorics of authentication and secrecy. J. Cryptol., 2, 23–49.
[317] Stinson, D. R. 1992. Combinatorial characterization of authentication codes. Design. Code. Cryptogr., 2, 175–187.
[318] Stinson, D. R. 2005. Cryptography, theory and practice. Third edn. Boca Raton, FL: Chapman & Hall/CRC.
[319] Sudan, M. 1997. Decoding of Reed Solomon codes beyond the errorcorrection bound. J. Complexity, 13(1), 180–193.
[320] Sugiyama, Y., Kasahara, M., Hirasawa, S. and Namekawa, T. 1975. A method for solving the key equation for decoding Goppa codes. Information and Control, 27, 87–99.
[321] Thas, J. A. 1992. M.D.S. codes and arcs in projective spaces: a survey Matematiche (Catania), 47(2), 315–328. Combinatorics 1992.
[322] Tsfasman, M. A., Vlădut,, S. G. and Zink, Th. 1982. Modular curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound. Math. Nachr., 109, 21–28.
[323] Tsfasman, M. A. and Vlădut,, S. G. 1991. Algebraic-geometric codes. Dordrecht: Kluwer Academic Publishers.
[324] Tsfasman, M. A. and Vlădut,, S. G. 1995. Geometric approach to higher weights. IEEE Trans. Inform. Theory, 41, 1564–1588.
[325] Tutte, W. T. 1947. A ring in graph theory. Proc. Amer. Math. Soc., 43, 26–40.
[326] Tutte, W. T. 1948. An algebraic theory of graphs. Ph.D. thesis, University of Cambridge.
[327] Tutte, W. T. 1954. A contribution to the theory of chromatic polynomials. Can. J. Math., 6, 80–91.
[328] Tutte, W.T. 1959. Matroids and graphs. Trans. Amer. Math. Soc., 90, 527–552.
[329] Tutte, W.T. 1965. Lectures on matroids. J. Res. Nat. Bur. Stand., Sect. B, 69, 1–47.
[330] Tutte, W. T. 1966. On the algebraic theory of graph coloring. J. Comb. Theory, 1, 15–50.
[331] Tutte, W. T. 1967. On dichromatic polynomials. J. Comb. Theory, 2, 301–320.
[332] Tutte, W. T. 1974. Cochromatic graphs. J. Comb. Theory, 16, 168–174.
[333] Tutte, W. T. 2004. Graphs-polynomials. Adv. Appl. Math., 32, 5–9.
[334] Valiant, L. G. 1979. Completeness classes in algebra. Pages 249–261 of: Conference record of the Eleventh Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1979). New York: ACM.
[335] Vardy, A. 1997. The intractability of computing the minimum distance of a code. IEEE Trans. Inform. Theory, 43, 1757–1766.
[336] Vardy, A. 1998. Codes, curves, and signals (Urbana, IL, 1997). Kluwer International Series in Engineering and Computer Science, vol. 485 Boston, MA: Kluwer Academic Publishers.
[337] Varshamov, R. R. 1957. Estimate of the number of signals in error correcting codes. Dokl. Acad. Nauk SSSR, 117, 739–741.
[338] Vlădut,, S. G. and Drinfeld, V. G. 1983. The number of points of an algebraic curve. Funktsional. Anal. i Prilozhen., 17(1), 68–69.
[339] Vlădut,, S. G. and Manin, Yu. I. 1984. Linear codes and modular curves. Pages 209–257 of: Current problems in mathematics, Vol. 25. Moscow: Itogi Nauki i Tekhniki. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform.
[340] Wegener, I. 1987. The complexity of Boolean functions. Wiley-Teubner Series in Computer Science. Chichester: John Wiley & Sons; Stuttgart: B. G. Teubner.
[341] Wegener, I. 2005. Complexity theory. Berlin: Springer-Verlag. Exploring the limits of efficient algorithms, Translated from the German by Randall Pruim.
[342] Wei, V. K. 1991. Generalized Hamming weights for linear codes. IEEE Trans. Inform. Theory, 37, 1412–1418.
[343] Welsh, D. J. A. 1976. Matroid theory. London: Academic Press.
[344] White, N. 1986. Theory of matroids. Cambridge: Encyclopedia of Mathmatics and its Applications, vol. 26, Cambridge University Press.
[345] White, N. 1992. Matroid applications. Cambridge: Encyclopedia of Mathmatics and its Applications, vol. 40, Cambridge University Press.
[346] Whitney, H. 1932a. The coloring of graphs. Ann. Math., 33, 688–718.
[347] Whitney, H. 1932b. A logical expansion in mathematics. Bull. Amer. Math. Soc., 38, 572–579.
[348] Whitney, H. 1935. On the abstract properties of linear dependence. Amer. J. Math., 57, 509–533.
[349] Whittle, G. 1995. A charactrization of the matroids representable over GF(3) and the rationals. J. Comb. Theory, Series B, 65, 222–261.
[350] Whittle, G. 1997. On matroids representable over GF(3) and other fields. Trans. Amer. Math. Soc., 349, 579–603.
[351] Wicker, S. B. 1998. Deep space applications. Pages 2119–2169 of: Handbook of coding theory, vol. 2 Amsterdam: North-Holland.
[352] Wicker, S. B. and Bhargava, V. K. 1994. Reed-Solomon codes and their applications. New York: IEEE Press.
[353] Wigderson, A. 2007. P, NP and mathematics –a computational complexity perspective. Pages 665–712 of: International Congress of Mathematicians. Vol. I. Zürich: European Mathematical Society.
[354] Wilson, R. J. and Watkins, J. J. 1990. Graphs; An introductory approach. New York: J. Wiley & Sons.
[355] Wirtz, M. 1988. On the parameters of Goppa codes. IEEE Trans. Inform. Theory, 34(5, part 2), 1341–1343. Coding techniques and coding theory.
[356] Wolf, J. K. and Elspas, B. 1963. Error-locating codes –a new concept in error control. IEEE Trans. Inform. Theory, IT-9, 113–117.
[357] Wozencraft, J. M. 1958. List decoding. Quarterly Progress Report, Research Laboratory of Electronics, MIT, 48, 90–95.
[358] Xing, C. and Chen, H. 2002. Improvements on parameters of one-point AG codes from Hermitian curves. IEEE Trans. Inform. Theory, 48(2), 535–537.
[359] Yang, K. and Kumar, P. V. 1992. On the true minimum distance of Hermitian codes. Pages 99–107 of: Coding theory and algebraic geometry (Luminy, 1991). Lecture Notes in Mathematics, vol. 1518 Berlin: Springer.
[360] Yao, A. C.-C. 1982. Protocols for Secure Computations. Pages 160–164 of: Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, FOCS 1982.
[361] Yuan, J. and Ding, C. 2006. Secret sharing schemes from three classes of linear codes. IEEE Trans. Inform. Theory, 52(1), 206–212.
[362] Zaslavsky, T. 1975. Facing up to arrangements: face-count fomulas for partitions of space by hyperplanes. Memoirs of the American Mathematical Society, no. 154. American Mathematical Society.
[363] Zaslavsky, T. 1982. Signed graph colouring. Discrete Math., 39, 215–228.
[364] Zhuangzi. 287 BC. Heaven and Earth, chapter 14. http://ctext.org/zhuangzi/heaven-and-earth.

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