Algebraico-geometric codes
Izvestiya. Mathematics , Tome 21 (1983) no. 1, pp. 75-91.

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An algebraico-geometric approach to coding theory is developed. Linear series on an algebraic curve are used for the construction and analysis of error-correcting codes. Bibliography: 27 titles.
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V. D. Goppa. Algebraico-geometric codes. Izvestiya. Mathematics , Tome 21 (1983) no. 1, pp. 75-91. http://geodesic.mathdoc.fr/item/IM2_1983_21_1_a2/

[1] Hamming R. W., “Error detecting and error-correcting codes”, Bell System Techn. J., 29:2 (1950), 147–160 | MR

[2] Dieudonné J., “The historical development of algebraic geometry”, The Amer. Math. Monthly, 72:8 (1972), 827–866 | DOI | MR

[3] Abhyankar S. S., “Historical ramblings in algebraic geometry and related algebra”, The Amer. Math. Monthly, 63:6 (1976), 409–448 | DOI | MR

[4] Severi F., Vorlesungen über algehraische geometry, Teubner, Leipzig, Berlin, 1921 | Zbl

[5] Segre B., Lectures on modern geometry, Roma, 1961 | MR

[6] Hocquenghem A., “Codes correcteurs d'erreurs”, Chiffres, 2 (1959), 147–156 | MR | Zbl

[7] Bose R., Chakravarti I., “Ermitian varieties in a finite projective space $PG(N, q^2)$”, Canad. J. Math., 18 (1966), 1161 | MR | Zbl

[8] Segre B., “Introduction to Galois geometry”, Atti Accad. Naz. Lincei Mem. CI. Sc. Fis. Mat. Natur., 8:5 (1967), 136–236 | MR

[9] Muller D. E., “Application of boolean algebra to switching circuit design and to error detecting”, IEEE Computers, 3:1 (1954), 6–12

[10] Gilbert E., “A comparison of signaling alphabets”, Beil Syst. Tech. J., 31:1 (1952), 504–522

[11] Varshamov R. R., “Otsenka chisla signalov v kodakh s korrektsiei oshibok”, Dokl. AN SSSR, 117:5 (1957), 739–741 | Zbl

[12] Kasami T., Tokura N., Ivadari E., Inagaki Ya., Teoriya kodirovaniya, Mir, M., 1978 | MR

[13] Fulton W., Algebraic curves. An introduction to algebraic geometry, Benjamin, N. Y., 1969 | MR | Zbl

[14] Schmidt W. M., Equations over finite fields. An elementarv approach, Springer, Berlin, 1976 | MR

[15] Leng S., Vvedenie v algebraicheskie i abelevy funktsii, Mir, M., 1976 | MR | Zbl

[16] Goppa V. D., “Ratsionalnoe predstavlenie kodov i $(L, g)$-kody”, Probl. pered. inform., 7:3 (1971), 41–49 | MR | Zbl

[17] Sugiyama Y., Kasahara M., Hirasawa S., Namekawa T., “Further results on Goppa codes and their applications to constructing efficient binary codes”, IEEE Trans. Inform. Theory, 22:5 (1976), 518–526 | DOI | MR | Zbl

[18] Mak-Vilyams Dzh., Sloen N., Teoriya kodov, ispravlyayuschikh oshibki, Svyaz, M., 1980

[19] Berlekamp E. R., Moreno O., “Extended double error-correcting binary Goppa codes are cyclic”, IEEE Trans. Inform. Theory, 19:6 (1973), 817–818 | DOI | MR | Zbl

[20] Moreno O., “Symmetries of binary Goppa codes”, IEEE Trans. Inform. Theory, 25:5 (1979), 609–612 | DOI | MR | Zbl

[21] Shannon C., “A mathematical theory of communication”, Bell Syst. Tech. J., 27 (1948), 379–423, 623–656 | MR

[22] Mahler K., Lectures on diophantine approximations. Part I. $g$-adic numbers and Roth's theorem, Univ. of Notre-Dame, 1961 | MR

[23] Piterson U., Kody, ispravlyayuschie oshibki, Mir, M., 1964 | MR

[24] Wiener N., The interpolation, extrapolation and smoothing of stationary time series, Wiley, N. Y., 1949 | Zbl

[25] Berlekemp E., Algebraicheskaya teoriya kodirovaniya, Mir, M., 1971 | MR

[26] Kokseter G. S. M., Mozer U. O. Dzh., Porozhdayuschie elementy i opredelyayuschie sootnosheniya diskretnykh grupp, Nauka, M., 1980

[27] Kasami T., “An upper bound on $k/n$ for af fine-invariant codes with fixed $d/n$”, IEEE Trans. Inform. Theory, 15:9 (1969), 174–176 | DOI | MR | Zbl