Perfect Codes In The Lp Metric

We investigate perfect codes in Zn in the ℓp metric. Upper bounds for the packing radius r of a linear perfect code, in terms of the metric parameter p and the dimension n are derived. For p=2 and n=2, 3, we determine all radii for which there exist linear perfect codes. The non-existence results for codes in Zn presented here imply non-existence results for codes over finite alphabets Zq, when the alphabet size is large enough, and have implications on some recent constructions of spherical codes. © 2015 Elsevier Ltd.537285Cohn, H., Elkies, N., New upper bounds on sphere packings I (2003) Ann. of Math., 157 (2), pp. 689-714Conway, J.H., Sloane, N.J.A., (1998) Sphere-Packings, Lattices, and Groups, , Springer-Verlag, New York, NY, USACosta, S.I.R., Campello, A., Jorge, G.C., Strapasson, J.E., Qureshi, C., Codes and lattices in the lp metric, in: Information Theory and Applications Workshop (2014), pp. 1-4. , ITA, FebruaryGolomb, S.W., (1996) Polyominoes: Puzzles, Patterns, Problems, and Packing, , Princeton Academic PressGolomb, S.W., Welch, L.R., Perfect codes in the lee metric and the packing of polyominoes (1970) SIAM J. Appl. Math., 18 (2), pp. 302-317Gruber, P.M., Lekkerkerker, C.G., (1987) Geometry of Numbers, , North-HollandHorak, P., AlBdaiwi, B.F., Diameter perfect lee codes (2012) IEEE Trans. Inform. Theory, 58 (8), pp. 5490-5499Horak, P., Grosek, O., A new approach towards the Golomb-Welch conjecture (2014) European J. Combin., 38, pp. 12-22Jorge, G.C., Campello, A., Costa, S.I.R., q-ary lattices in the ℓp norm and a generalization of the Lee metric (2013) Workshop on Coding and Cryptography, pp. 1-8Kovacevic, M., Difference sets and codes in $ A_n $ lattices September 2014. ArXiv e-printsLekkerkerker, C.G., (1969) Geometry of Numbers, , Wolters-NoordhoffNiven, I., Zuckerman, H.S., Montgomery, H.L., (1991) An Introduction to The Theory of Numbers, , WileyRush, J.A., Sloane, N.J.A., An improvement to the Minkowski-Hlawka bound for packing superballs (1987) Mathematika, 34, pp. 8-18Schwartz, M., Quasi-cross lattice tilings with applications to flash memory (2012) IEEE Trans. Inform. Theory, 58 (4), pp. 2397-2405Schwartz, M., On the non-existence of lattice tilings by quasi-crosses (2014) European J. Combin., 36, pp. 130-142Solé, P., Belfiore, J.-C., Constructive spherical codes near the Shannon bound (2013) Des. Codes Cryptogr., 66 (1-3), pp. 17-26Stein, S.K., Szabo, S., (1994) Algebra and Tiling: Homomorphisms in the Service of Geometry, , The Mathematical Association of AmericaVaughan, R.C., Wooley, T.D., Waring's problem: a survey, in: Number Theory for the Millennium (2012), pp. 301-340. , III, Urbana, IL, 200