# Sphere Packings, Lattices and Groups

Springer Science & Business Media, Dec 7, 1998 - Mathematics - 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.

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### Contents

 IV 1 V 3 VI 7 VII 8 VIII 12 IX 21 X 24 XI 26
 CLII 286 CLIII 287 CLIV 290 CLV 292 CLVI 293 CLVII 294 CLIX 295 CLX 296

 XIII 27 XIV 29 XV 31 XVI 33 XVII 36 XVIII 40 XIX 41 XX 42 XXI 44 XXII 47 XXIII 50 XXIV 52 XXV 56 XXVI 59 XXVIII 63 XXIX 66 XXX 69 XXXI 71 XXXII 75 XXXIII 77 XXXIV 79 XXXVI 81 XXXVII 82 XXXVIII 83 XXXIX 84 XL 85 XLI 86 XLII 87 XLIII 88 XLIV 89 XLV 90 XLVI 92 XLVII 94 XLVIII 95 XLIX 99 L 101 LI 102 LII 106 LIII 108 LIV 110 LV 112 LVI 113 LVIII 115 LIX 116 LX 117 LXI 118 LXII 119 LXIII 120 LXV 124 LXVI 125 LXVII 127 LXVIII 129 LXIX 131 LXX 136 LXXI 137 LXXIII 138 LXXVI 139 LXXVII 140 LXXVIII 141 LXXIX 142 LXXXII 144 LXXXIV 145 LXXXVI 146 LXXXVII 147 LXXXVIII 148 LXXXIX 149 XC 150 XCI 151 XCIII 152 XCIV 153 XCV 155 XCVI 157 XCVII 163 XCVIII 168 XCIX 170 C 174 CI 176 CII 177 CIII 179 CIV 181 CV 182 CVI 185 CVII 189 CVIII 191 CX 193 CXI 197 CXII 202 CXIII 205 CXIV 206 CXV 207 CXVI 210 CXVII 211 CXVIII 215 CXIX 221 CXX 222 CXXI 224 CXXII 227 CXXIV 229 CXXV 232 CXXVI 233 CXXVII 235 CXXVIII 236 CXXIX 238 CXXX 245 CXXXI 249 CXXXII 250 CXXXIII 252 CXXXIV 253 CXXXV 256 CXXXVI 257 CXXXVII 258 CXXXVIII 260 CXXXIX 263 CXL 265 CXLI 267 CXLII 269 CXLIII 271 CXLIV 273 CXLVI 274 CXLVII 276 CXLVIII 278 CL 279 CLI 283
 CLXI 299 CLXII 300 CLXIII 302 CLXV 303 CLXVI 305 CLXVII 307 CLXVIII 308 CLXIX 309 CLXX 311 CLXXI 314 CLXXII 316 CLXXIII 318 CLXXIV 319 CLXXVI 320 CLXXVII 323 CLXXVIII 327 CLXXIX 331 CLXXX 337 CLXXXI 338 CLXXXII 340 CLXXXIII 342 CLXXXIV 344 CLXXXV 345 CLXXXVI 349 CLXXXVII 352 CLXXXVIII 354 CLXXXIX 355 CXC 356 CXCII 357 CXCIII 359 CXCIV 364 CXCV 366 CXCVII 367 CXCIX 368 CC 369 CCI 370 CCIII 372 CCIV 373 CCVII 375 CCVIII 377 CCIX 378 CCXI 379 CCXII 380 CCXIV 381 CCXV 382 CCXVII 384 CCXVIII 385 CCXIX 386 CCXX 388 CCXXI 389 CCXXII 390 CCXXIV 391 CCXXV 392 CCXXVI 393 CCXXVII 396 CCXXVIII 399 CCXXX 402 CCXXXI 406 CCXXXII 408 CCXXXIII 410 CCXXXIV 413 CCXXXV 421 CCXXXVI 429 CCXXXVII 430 CCXXXVIII 433 CCXXXIX 436 CCXL 439 CCXLI 441 CCXLII 443 CCXLIII 445 CCXLIV 446 CCXLV 448 CCXLVI 449 CCXLVII 450 CCXLIX 451 CCL 453 CCLII 454 CCLV 455 CCLVII 456 CCLIX 457 CCLXII 458 CCLXIII 461 CCLXIV 464 CCLXVI 465 CCLXVII 474 CCLXVIII 476 CCLXIX 478 CCLXX 480 CCLXXI 482 CCLXXII 486 CCLXXIII 497 CCLXXIV 504 CCLXXV 508 CCLXXVI 512 CCLXXVII 515 CCLXXVIII 516 CCLXXIX 521 CCLXXX 524 CCLXXXI 525 CCLXXXII 529 CCLXXXIII 530 CCLXXXIV 534 CCLXXXV 543 CCLXXXVI 549 CCLXXXVIII 552 CCLXXXIX 556 CCXC 558 CCXCIII 559 CCXCVI 560 CCXCVII 561 CCXCIX 562 CCC 563 CCCII 564 CCCIII 565 CCCV 566 CCCVI 568 CCCVII 569 CCCVIII 570 CCCIX 574 CCCX 642 CCCXI 681 Copyright

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