Sphere Packings, Lattices and GroupsSpringer Science & Business Media, 7 déc. 1998 - 706 pages We now apply the algorithm above to find the 121 orbits of norm -2 vectors from the (known) nann 0 vectors, and then apply it again to find the 665 orbits of nann -4 vectors from the vectors of nann 0 and -2. The neighbors of a strictly 24 dimensional odd unimodular lattice can be found as follows. If a norm -4 vector v E II . corresponds to the sum 25 1 of a strictly 24 dimensional odd unimodular lattice A and a !-dimensional lattice, then there are exactly two nonn-0 vectors of ll25,1 having inner product -2 with v, and these nann 0 vectors correspond to the two even neighbors of A. The enumeration of the odd 24-dimensional lattices. Figure 17.1 shows the neighborhood graph for the Niemeier lattices, which has a node for each Niemeier lattice. If A and B are neighboring Niemeier lattices, there are three integral lattices containing A n B, namely A, B, and an odd unimodular lattice C (cf. [Kne4]). An edge is drawn between nodes A and B in Fig. 17.1 for each strictly 24-dimensional unimodular lattice arising in this way. Thus there is a one-to-one correspondence between the strictly 24-dimensional odd unimodular lattices and the edges of our neighborhood graph. The 156 lattices are shown in Table 17 .I. Figure I 7. I also shows the corresponding graphs for dimensions 8 and 16. |
Table des matières
IV | lxxiii |
VI | 1 |
VII | 5 |
VIII | 6 |
IX | 8 |
X | 15 |
XII | 18 |
XIII | 20 |
CLXXXVIII | 280 |
CLXXXIX | 281 |
CXC | 284 |
CXCI | 286 |
CXCII | 287 |
CXCIII | 288 |
CXCV | 289 |
CXCVI | 290 |
XIV | 21 |
XV | 23 |
XVI | 25 |
XVIII | 27 |
XIX | 30 |
XX | 34 |
XXI | 35 |
XXIII | 36 |
XXIV | 38 |
XXV | 41 |
XXVI | 44 |
XXVII | 46 |
XXVIII | 50 |
XXIX | 53 |
XXXI | 57 |
XXXV | 60 |
XXXVI | 63 |
XXXVII | 65 |
XXXVIII | 69 |
XL | 71 |
XLI | 73 |
XLIII | 75 |
XLIV | 76 |
XLV | 77 |
XLVI | 78 |
XLVII | 79 |
XLVIII | 80 |
XLIX | 81 |
LII | 82 |
LIV | 83 |
LV | 84 |
LVI | 86 |
LVII | 88 |
LIX | 89 |
LX | 93 |
LXI | 95 |
LXII | 96 |
LXIII | 100 |
LXIV | 102 |
LXV | 104 |
LXVI | 106 |
LXVII | 107 |
LXIX | 109 |
LXX | 110 |
LXXI | 111 |
LXXII | 112 |
LXXIII | 113 |
LXXIV | 114 |
LXXV | 118 |
LXXVI | 119 |
LXXVII | 121 |
LXXVIII | 123 |
LXXIX | 125 |
LXXX | 130 |
LXXXI | 131 |
LXXXV | 132 |
LXXXVIII | 133 |
LXXXIX | 134 |
XC | 135 |
XCII | 136 |
XCV | 138 |
XCVII | 139 |
XCVIII | 140 |
C | 141 |
CI | 142 |
CIII | 143 |
CV | 144 |
CVIII | 145 |
CX | 146 |
CXI | 147 |
CXII | 149 |
CXIII | 151 |
CXVI | 157 |
CXVII | 162 |
CXVIII | 164 |
CXIX | 168 |
CXX | 170 |
CXXI | 171 |
CXXII | 173 |
CXXIII | 175 |
CXXVII | 176 |
CXXVIII | 179 |
CXXIX | 183 |
CXXX | 185 |
CXXXII | 187 |
CXXXIII | 191 |
CXXXIV | 196 |
CXXXV | 199 |
CXXXVI | 200 |
CXXXIX | 201 |
CXLI | 204 |
CXLII | 205 |
CXLIII | 209 |
CXLIV | 215 |
CXLV | 216 |
CXLVI | 218 |
CXLIX | 221 |
CLI | 223 |
CLII | 226 |
CLIV | 227 |
CLV | 229 |
CLVI | 230 |
CLVII | 232 |
CLVIII | 239 |
CLX | 243 |
CLXI | 244 |
CLXII | 246 |
CLXIII | 247 |
CLXIV | 250 |
CLXV | 251 |
CLXVII | 252 |
CLXVIII | 254 |
CLXIX | 257 |
CLXX | 259 |
CLXXI | 261 |
CLXXVI | 263 |
CLXXIX | 265 |
CLXXX | 267 |
CLXXXII | 268 |
CLXXXIV | 270 |
CLXXXV | 272 |
CLXXXVI | 273 |
CLXXXVII | 277 |
CXCVII | 293 |
CC | 294 |
CCI | 296 |
CCIII | 297 |
CCIV | 299 |
CCV | 301 |
CCVI | 302 |
CCVII | 303 |
CCVIII | 305 |
CCIX | 308 |
CCX | 310 |
CCXI | 312 |
CCXII | 313 |
CCXIII | 314 |
CCXIV | 317 |
CCXV | 321 |
CCXVI | 325 |
CCXIX | 331 |
CCXXII | 332 |
CCXXIII | 334 |
CCXXVII | 336 |
CCXXVIII | 338 |
CCXXIX | 339 |
CCXXX | 343 |
CCXXXI | 346 |
CCXXXV | 348 |
CCXXXVII | 349 |
CCXXXVIII | 350 |
CCXL | 351 |
CCXLI | 353 |
CCXLII | 358 |
CCXLIII | 360 |
CCXLV | 361 |
CCXLVII | 362 |
CCXLVIII | 363 |
CCXLIX | 364 |
CCLI | 366 |
CCLII | 367 |
CCLIII | 369 |
CCLIV | 371 |
CCLV | 372 |
CCLVI | 373 |
CCLVII | 374 |
CCLIX | 375 |
CCLX | 376 |
CCLXII | 378 |
CCLXIII | 379 |
CCLXV | 380 |
CCLXVI | 382 |
CCLXVIII | 383 |
CCLXIX | 384 |
CCLXXI | 385 |
CCLXXII | 386 |
CCLXXIII | 387 |
CCLXXIV | 390 |
CCLXXVI | 393 |
CCLXXVIII | 396 |
CCLXXIX | 400 |
CCLXXXIII | 402 |
CCLXXXIV | 402 |
CCLXXXV | 405 |
CCLXXXVI | 413 |
CCXC | 421 |
CCXCI | 422 |
CCXCII | 425 |
CCXCIII | 428 |
CCXCIV | 431 |
CCXCV | 433 |
CCXCIX | 435 |
CCC | 437 |
CCCIV | 438 |
CCCV | 440 |
CCCVI | 441 |
CCCVII | 442 |
CCCIX | 443 |
CCCXIII | 445 |
CCCXV | 446 |
CCCXVIII | 447 |
CCCXX | 448 |
CCCXXI | 449 |
CCCXXIV | 450 |
CCCXXVI | 453 |
CCCXXVII | 456 |
CCCXXIX | 457 |
CCCXXX | 466 |
CCCXXXI | 468 |
CCCXXXII | 470 |
CCCXXXV | 472 |
CCCXXXVIII | 474 |
CCCXXXIX | 478 |
CCCXL | 489 |
CCCXLI | 496 |
CCCXLII | 500 |
CCCXLIV | 504 |
CCCXLV | 507 |
CCCXLVII | 508 |
CCCXLVIII | 513 |
CCCXLIX | 516 |
CCCLI | 517 |
CCCLII | 521 |
CCCLV | 522 |
CCCLVI | 526 |
CCCLX | 535 |
CCCLXI | 541 |
CCCLXIII | 544 |
CCCLXIV | 548 |
CCCLXVIII | 550 |
CCCLXX | 551 |
CCCLXXII | 552 |
CCCLXXIII | 553 |
CCCLXXV | 554 |
CCCLXXVI | 555 |
CCCLXXVIII | 556 |
CCCLXXIX | 557 |
CCCLXXXI | 558 |
CCCLXXXII | 560 |
CCCLXXXIII | 561 |
CCCLXXXIV | 562 |
CCCLXXXVII | 566 |
CCCLXXXVIII | 634 |
673 | |
Autres éditions - Tout afficher
Expressions et termes fréquents
algebraic algorithm automorphism group binary C-set center density Chap chapter code of length codewords compute congruent construction contains coordinates corresponding covering radius Coxeter Coxeter-Dynkin diagram deep hole defined denote densest determinant digits dimensions dual elements equivalent Euclidean example finite follows function genus given glue vectors Golay code inner product invariant isomorphic kissing number laminated lattices lattice packing lattice points Leech lattice Leech roots linear log2 Mac6 Math Mathieu group matrix maximal minimal distance minimal norm minimal vectors modular modulo n-dimensional N. J. A. Sloane Niemeier lattices nodes nonlattice packings nonzero obtained octad orthogonal p-adic permutation PGIT polynomial polytope problem proof quadratic forms quantizer reflection groups root lattices root system self-dual codes simple group space sphere packings spherical codes spinor subgroup t-designs Table tetrads Theorem theta series unimodular lattices unique Voronoi cell weight enumerator
Fréquemment cités
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