Bibliography. [1] Allenby R. B. J. T., Rings, Fields and Groups. An introduction to abstract algebra, London, Arnold, 1986.

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1 Bibliography [1] Allenby R. B. J. T., Rings, Fields and Groups. An introduction to abstract algebra, London, Arnold, [2] Artin E., Galois Theory, second edition, Notre Dame Mathematical Lectures, Number 2, Indiana, [3] Artin M., Algebra, Englewood Cliff, Prentice Hall, [4] Assmus E. F. Jr, Key J. D., Designs and their codes, Cambridge, Cambridge Univ. Press, [5] Atiyah M. F. and Macdonald I. G., Introduction to Commutative Algebra, Reading, Massachusetts - Menlo Park, California, Addison - Wesley Publishing Co., [6] Baker R D., van Lint J. H. and Wilson R M., On the Preparata codes and Goethals codes, IEEE Trans. Inform. Theory, 29 (1983), [7] Bastida J. R, Field Extensions and Galois Theory, Reading Massachusetts, Addison-Wesley, [8] Blake I. F., Codes over integer residue rings, Inform. Control, 29 (1975), [9] Bonnecaze A., Sole P., and Calderbank A. R, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory, 41 (1995), [10] Calderbank A. R, Cameron P. J., Kantor W. M., Seidel J. J., Z4- Kerdock codes, orthogonal spreads, and extremal euclidean linesets, Proc. London Math. Soc., 75 (1997),

2 168 BIBLIOGRAPHY [11] Calderbank A. R, Hammons A. R Jr., Kumar P. V., Sloane N. J. A. and Sole' P., The Z4 - linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), [12] Calderbank A. R, McGuire G., Z4-linear codes obtained as projections of Kerdock and Delsarte-Goethals codes, Lin. Alg. and Appl., (1995), [13] Calderbank A. R, Sloane N. J. A., Modular and p-adic cyclic codes, Des., Codes and Cryptogr., 6 (1995), [14] Cameron P. J., van Lint J. H., Designs, Graphs, Codes and their links, London Math. Soc. Student Texts Vol. 22, Cambridge, Cambridge University Press, [15] Carlet C., A simple description of Kerdock codes, Lecture Notes in Computer Science, 388 (1989), [16] Carlet C., On Z4-Duality, IEEE Trans. Inform. Theory, 41 (1995), [17] Cohn P. M., Algebra, vol. II, London-New York-Sydney, J. Wiley and Sons, [18] Conway J. H., Sloane N. J. A., Self-dual codes over the integers modulo 4, J. Comb. Theory, Ser. A, 62 (1993), [19] Davenport J., Siret Y. and Tournier E., Computer Algebra: systems and algorithms for algebraic computations, London, Academic Press, [20] Davis J. A., Jedwab J, and Paterson G., Codes, Correlations and power control in OFDM, in Difference Sets, Sequences and their correlation properties, , eds. Pott A., Kumar P. V. et al., Kluwer Academic Publishers, Boston, [21] Dieudonne J., La geometrie des groupes classiques finis, Berlin Gottingen-Heidelberg, Springer, [22] Edwards H. M., Galois Theory, New York, Springer, [23] Eisenbud D., Commutative Algebra, with a view toward Algebraic Geometry, Springer, Berlin, [24] Ellis G., Rings and Fields, Oxford, Clarendon Press, 1992.

3 BIBLIOGRAPHY 169 [25) Ganesan N., Properties of rings with a Finite Number of Zero Divisors II, Math. Annalen, 161 (1965), [26) Gilmer R., R-automorphism of R[x), Proc. London Math. Soc., 18 (1968), [27) Goethals J. M., Snover S. L., Nearly perfect binary codes, Disc. Math., 3 (1972), [28) Goldstein L. J., Abstract Algebra: a first course, Englewood Cliff, Prentice Hall, [29) Hamming R. W., Error detecting and error correcting codes, Bell Syst. Tech. J., 29 (1950), [30) Hartshorne J., Algebraic Geometry, Springer - Verlag, GTM 52, [31) Herstein 1. N., Topics in Algebra, New York - Toronto - London, Blaisdell Pubbl. Co., [32) Herstein I. N., Abstract Algebra, New York, Mac Millan, [33) Hirschfeld J. W. P., Projective geometries over finite fields. Second edition, Oxford, Claredon Press, [34) Hoffman D. G., Leonard D. A. et al., Coding Theory: The essentials, New York, Marcel Dekker, [35] Hungerford T. W., Algebra, New York-Heidelberg, Springer-Verlag, [36) Jacobson N., Lectures in Abstract Algebra, vol. III, New York-Toronto-London-Melbourne, American Book-Van Nostrand Reinhold, [37) Jacobson N., Basic Algebra I and II, Second Edition, New York, W. H. Freeman and Co., [38] Janusz G., Separable Algebras over Commutative Rings, Trans. A.M.S., 122 (1966), [39) Jungnickel D., Finite Fields: structure and arithmetics, Mannheim, BI-Wiss.-Verl., 1993.

4 170 BIBLIOGRAPHY [40J Kantor W. M., Spreads, translation planes, and Kerdock sets I, SIAM, J. Alg. Discr. Math., 3 (1982), [41J Kantor W. M., Spreads, translation planes, and Kerdock sets II, SIAM, J. Alg. Discr. Math., 3 (1982), [42J Kantor W. M., An exponential number of generalized Kerdock codes, Inform. Control, 53 (1982), [43J Karpilovsky G., Unit Groups of Classical Rings, Oxford, Claredon, [44J Karpilovsky G., Topics in Field Theory, Amsterdam, North Holland, [45J Kerdock A. M., A class of low rate non linear binary codes, Inform. Control, 20 (1972), [46J Klingenberg W., Projective und affine Ebene mit Nachbarelementen, Math. Z., 60 (1960), [47J Krull W., Algebraische Theorie der Ringe, Math. Ann., 92 (1924), [48J Lang S., Algebraic Number Theory, New York, Springer, [49J Lang S., Algebra, Third Ed., Reading-Massachusetts, Addison Wesley, [50J Lidl R., Niederreiter H., Finite Fields, Readings Massachusetts, Addison-Wesley, [51] Lidl R., Niederreiter H., Introduction to Finite Fields and Their Applications, Cambridge, Cambridge University Press, [52] Machi' A., Algebra per il Calcolo Simbolico, Roma, Edizioni Kappa, [53] MacWilliams F. J., Sloane N. J. A., The Theory of error-correcting codes, Amsterdam, North Holland, [54] Malliavin M. - P., Algebre Commutative. Applications en Geometrie et TMorie des Nombres, Paris, Masson, [55] Matsumura H., Commutative Algebra, Readings Massachusetts, B. Cummings Publishing Program, 1980.

5 BIBLIOGRAPHY 171 [56J Mc Donald B. R, Finite Rings with Identity, Inc. New York, Marcell Dekker, [57J Mc Eliece R J., Finite fields for computer scientists and engineers, Boston, Kluwer Accademic Publ., [58] Motzkin T., The Euclidean Algorithm, Bull. Amer. Math. Soc., 55 (1949), [59J Niven 1. and Zuckerman H. S., An Introduction to the Theory of Numbers. Fourth edition., New York, Wiley, [60] Pless V., Qian Z., Cyclic Codes and Quadratic Residue Codes over Z4, IEEE Trans. Inform. Theory, 42 (1996), [61] Preparata F. P., A class of optimum non linear double-error correcting codes, Inform. Control, 13 (1968), [62J Pott A., Kumar P. V., Helleseth T., and Jungnickel D., Proceedings of the NATO Advanced Study Institute on Difference Sets, Sequences and their Correlation Properties, Bad Winsheim, 2-14 August 1998, NATO Science Series C - 542, Dordrecht, Kluwer Academic Publishers, [63J Raghavendran R, Finite Associative Rings, Compositio Math., 21 (1969), [64] Roman S., Field Theory, New York, Springer, [65] Rose J. S., A Course on Group Theory, Cambridge, Cambridge University Press, [66] Sharp R Y., Steps in Commutative Algebra, Cambridge, Cambridge University Press, [67] Snover S. L., The uniqueness of the Nordstrom-Robinson and Golay binary codes, Ph.D. Thesis, Dept. of Mathematics, Michigan State Univ., [68J Stewart 1., Galois Theory, second edition, London, Chapman and Hall, [69] van Lint J. H., Introduction to Coding Theory, third edition, New York, Springer, 1998.

6 172 BIBLIOGRAPHY [70] Weil A., Basic Number Theory, second edition, Berlin-Heidelberg New York, Springer-Verlag, [71] Zariski O. and Samuel P., Commutative Algebra (vol I and II), Princeton, Van Nostrand, 1958 and 1960.

7 Index algebraic element 15 algebraically closed field 17 algebraic closure 17 algebraic integer 13 algebraic number 13 Artinian ring 7 associates, elements 12 automorphism 35 b-adic 21 basic irreducible polynomial 53 Boolean function 129 character 137 characteristic ideal 54 characteristic, of a ring 3 code (over a finite field) binary binary Kerdock classical Preparata cyclic distance invariant dual- 123 extended Preparata formal dual- 134 Generalized Reed-Muller generating function of Hamming linear (m, M, d) [m, k, dj minimal- 127 nearly-perfect nonlinear Nordstrom-Robinson projective q-ary Reed-Muller self-dual 123 ternary code (over a finite ring) binary image of a cyclic dual linear linear quaternary of type n ~ p(n-i)ki l, quaternary Kerdock RM2k (r, m) Z4-linear Z4-Preparata ZRM2k(r,m) coefficient ring 106 commutator 86 conjugate, of an element 36 Dedekind domain 83 degree, of an algebraic element 15 degree, of an extension 16 derived group 86 derived series 86 distance distribution, - 134

8 174 INDEX Hamming Lee minimum minimum Lee division ring 37 divisor 12 enumerator complete weight distance Hamming weight - 131, 155 Lee weight symmetrized weight weight - of an element 131 equivalent codes 121 Euclidean domain 11 Euclidean field 13 Euler function 44 extraspecial group 161 field 2 finite field 29 finite chain ring 119 formal derivative 56 formal power series 7 fractional ideal 83 Frobenius automorphism 36 fundamental subfield 3 fundamental subring 3 Fundamental Thm. of Algebra 17 Thm. of Arithmetic 7, 43 Galois correspondence theorem 77 Galois extension 62, 72 Galois field 2, 17 Galois ring 7, 20, 27, 45, 82 Gauss integers 12 Gauss lemma 19 g.c.d. 13 generalized Boolean function 147 generalized Hensel's lemma 49 ghost component 90 Gray map 155 group algebra 130 Hensel's lemma, integral version 22 Hilbert Basis Theorem 18 Homomorphism theorem 2 idempotent element 41 integral domain 1 invertible element 1 irreducible element 12, 21 Jacobson radical 8 Kerdock set 136 K-morphism 62 Krull dimension 6, 18 I.c.m 13 lift (of a vector) 163 localization 6 local morphism 6 local polynomial 69 local ring 6 MacWilliams' Identity, transform, matrix generator - (over a field) 122 generator - (over a ring) 147 parity check - (over a field) 123 parity check - (over a ring) 147 maximal ideal 4 maximal Spectrum, of a ring 5 minimal polynomial 15

9 INDEX m-th power of an ideal 40 multiple 12 multiplicative system 6 Nakayama's lemma 66 Newton binomial formula 96 nilpotency class 10 nilpotent 1 nilradical 8 Noetherian ring 18 n-th cyclotomic field 44 n-th derived group 86 n-th root of unity 44 one-group 116 orthogonal elements 41 p-adic 21, 83 perfect field 63 permutation equivalence 145 polynomial ring 9 polynomial (related to codes) generator parity check primitive idempotent primary ideal 47 primary polynomial 47 primary ring 110 prime element 12 prime field 3 prime ideal 4 prime ring 3 primitive element 30 element, Theorem of - 67 polynomial 26, 33 polynomial (Gauss def.) 19 P.J.D. 11 principal ideal ring 11 principal unit 116 projective module 84 proper divisor 58 proper ideal 3 quadratic field 13 Quasi-Galois ring 81, 107 quaternary Kerdock set 162 quotient field radical ideal 8 radical, of an ideal 8 R-automorphism 71 regular polynomial 21, 47 relatively prime ideals 39 relatively prime polynomials 47 residue field 6 ring extension 65 semi local ring 7 separable element 62, 75 separable extension 62, 66 separable polynomial 62, 69 skew-field 37 simple extension (of a field) 16 simple extension (of a ring) 65 simple ring 4 solvable group 86 space symplectic n+(2n, q) Specm(R) 5 Spec(R) 5 Spectrum, of a ring 5 splitting field 31, 62 splitting ring 77 spread equivalence 138 orthogonal symplectic strongly separable algebra 84 subspace totally singular - 137

10 176 INDEX totally isotropic Teichmiiller set 83 transcendent element 9 U.F.D.13 unique factorization theorem 19 unit 1, 12, 21 unramified extension 66 valuation 11 valuation morphism 15 Wedderburn's theorem 37 weight distribution, Lee minimum minimum Lee Wilson's theorem 32 Witt vectors 89 word (of an alphabet) 121 Z4-duall51 Z2n-duality 158 Z2n-linearity 158 zero-divisor 1 Zorn's lemma 5

Extended Index. 89f depth (of a prime ideal) 121f Artin-Rees Lemma. 107f descending chain condition 74f Artinian module

Extended Index. 89f depth (of a prime ideal) 121f Artin-Rees Lemma. 107f descending chain condition 74f Artinian module Extended Index cokernel 19f for Atiyah and MacDonald's Introduction to Commutative Algebra colon operator 8f Key: comaximal ideals 7f - listings ending in f give the page where the term is defined commutative