Error-correcting codes and Minkowski’s conjecture

[1] ASTOLA, J.: On perfect codes in the Lee metric, Ann. Univ. Turku, Ser. A I 176 (1978), 56 p.Search in Google Scholar

[2] AL-BDAIWI, B. F.-BOSE, B.: Quasi-perfect Lee distance codes, IEEE Tran. Inform. Theory 49 (2003), 1535-1539.10.1109/TIT.2003.811922Search in Google Scholar

[3] AL-BDAIWI, B. F.-HORAK, P.-MILLAZO, L.: Perfect 1-error correcting Lee codes, Des. Codes Cryptogr. 52 (2009), 155-162.10.1007/s10623-009-9273-3Search in Google Scholar

[4] GOLOMB, S.W.-WELSH, L. R.: Algebraic coding and the Lee metric, in: Error Correct. Codes, Wiley, New York, 1968, pp. 175-189.Search in Google Scholar

[5] GRAVIER, S.-MOLLARD, M.-PAYAN, CH.: On the non-existence of 3-dimensionaltiling in the Lee metric, European J. Combin. 19 (1998), 567-572.10.1006/eujc.1998.0211Search in Google Scholar

[6] GRAVIER, S.-MOLLARD, M.-PAYAN, CH.: On the nonexistence of three--dimensional tiling in the Lee metric. II. Discrete Math. 235 (2001), 151-157.Search in Google Scholar

[7] HAJÓS, G.: ¨Uber eifache und merfache Bedeckung des n-dimensional Raumes mit einemW¨urfelgitter, Math. Zeit. 47 (1942), 427-467.10.1007/BF01180974Search in Google Scholar

[8] HORAK, P.-AL-BDAIWI, B. F.: Fast decoding quasi-perfect Lee distance codes, Des. Codes Cryptogr. 40 (2006), 357-367.10.1007/s10623-006-0025-3Search in Google Scholar

[9] HORAK, P.: Tiling in Lee metric, European J. Combin. 30 (2009) 480-489.10.1016/j.ejc.2008.04.007Search in Google Scholar

[10] HORAK, P.: On perfect Lee codes, Discrete Math. 309 (2009), 5551-5561.10.1016/j.disc.2008.03.019Search in Google Scholar

[11] HORAK, P.-AL-BDAIWI, B. F.: The number of tilings of Rn by crosses, submitted.Search in Google Scholar

[12] KÁRTESZI, F.: Szemléletes geometria. Gondolat, Budapest, 1966.Search in Google Scholar

[13] MOLNÁR, E.: Sui mosaici dello spazio di dimensione n, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 51 (1971), 177-185.Search in Google Scholar

[14] KELLER, O. H.: Über die lückenlose Einfüllung des Raumes mit Würfeln, J. Reine Angew. Math. 177 (1930), 231-248.10.1515/crll.1930.163.231Search in Google Scholar

[15] LAGARIAS, J. F.-SHOR, P. W.: Keller’s cube-tiling conjecture is false in high dimensions, Bull. Amer. Math. Soc. 27 (1992), 279-283.10.1090/S0273-0979-1992-00318-XSearch in Google Scholar

[16] MACKEY, J.: A cube tiling of dimension eight with no facesharing, Discrete Comput. Geom. 28 (2002) 275-279.Search in Google Scholar

[17] PERRON, O.: Modulartige lückenlose Ausfüllungdes Rn mit kongruenten Würfeln I, II, Math. Ann. 117 (1940), 415-447; 117 (1941), 609-658.Search in Google Scholar

[18] POST, K. A.: Nonexistence theorem on perfect Lee codes over large alphabets, Inform. and Control 29 (1975), 369-380.10.1016/S0019-9958(75)80005-2Search in Google Scholar

[19] STEIN, S. K.: Factoring by subsets, Pacific J. Math. 22 (1967), 523-541.10.2140/pjm.1967.22.523Search in Google Scholar

[20] STEIN, S. K.-SZABÓ, S.: Algebra and Tilings: Homomorphisms in the Service of Geometry. Carus Mathematical Monographs, Vol. 25 Math. Assoc. Amer., Washington, DC, 1994.10.5948/UPO9781614440246Search in Google Scholar

[21] SZABÓ, S.: On mosaics consisting of multidimensional crosses, Acta Math. Acad. Sci. Hung. 38 (1981), 191-203.10.1007/BF01917533Search in Google Scholar

[22] SZABÓ, S.: A star polyhedron that tiles but not as fundamental domain, Colloq. Math. Soc. Janos Bolyai 48 (1985), 531-544.Search in Google Scholar

[23] ŚPACAPAN, S.: Nonexistence of face-to-face four-dimensional tilings in the Lee metric, European. J. Combin. 28 (2007) 127-133.Search in Google Scholar

[24] ULRICH,W.: Non-binary error-correction codes, Kibern. Sb. 7 (1963), 7-59; transl. from BEll System Tech. J. 36 (1957), 1341-1387.Search in Google Scholar