Best Known (99−85, 99, s)-Nets in Base 27
(99−85, 99, 96)-Net over F27 — Constructive and digital
Digital (14, 99, 96)-net over F27, using
- t-expansion [i] based on digital (11, 99, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 11 and N(F) ≥ 96, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
(99−85, 99, 136)-Net over F27 — Digital
Digital (14, 99, 136)-net over F27, using
- t-expansion [i] based on digital (13, 99, 136)-net over F27, using
- net from sequence [i] based on digital (13, 135)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 13 and N(F) ≥ 136, using
- net from sequence [i] based on digital (13, 135)-sequence over F27, using
(99−85, 99, 1364)-Net over F27 — Upper bound on s (digital)
There is no digital (14, 99, 1365)-net over F27, because
- 1 times m-reduction [i] would yield digital (14, 98, 1365)-net over F27, but
- extracting embedded orthogonal array [i] would yield linear OA(2798, 1365, F27, 84) (dual of [1365, 1267, 85]-code), but
- the Johnson bound shows that N ≤ 34 452087 196600 891384 661097 492071 182859 453705 127207 185030 439948 930744 421510 095870 698466 631039 950462 964115 245839 028145 831785 112437 744595 806396 861459 024925 590921 508030 674921 720034 998367 649394 359536 767163 417944 648454 497200 149601 185006 122673 930472 722184 205450 766672 853595 895543 549450 557003 505699 468604 430817 537172 951436 947424 156552 916044 939361 134104 917225 297118 472946 001179 907498 514811 682789 773551 180685 331800 689044 864647 896179 762994 807297 886617 233227 803486 540053 996414 545901 823077 344639 634276 542848 283281 432856 858035 797838 175312 401991 682847 047150 324363 584627 400344 667934 645502 974103 465907 792654 427630 995736 509263 511381 152533 598160 587861 738886 131083 787485 784965 257500 535455 297455 046725 127590 567216 534717 487299 271927 731973 714682 536053 808679 252587 846914 399848 188163 654093 572180 634063 349862 803504 334354 924265 729204 817930 610005 757678 753506 045073 075826 031156 060421 018380 982776 607471 062566 504223 671995 419542 189818 060404 892700 390030 659926 124698 325949 944302 861659 846438 849472 016692 021655 678537 264689 378264 508963 188721 592273 458334 771935 840135 624540 565539 766860 482182 067722 447517 190744 732973 335770 396594 308208 394623 866701 793548 131061 830891 995580 170666 893026 617442 277146 876405 383305 273900 523362 065205 289357 163664 893677 593756 182822 095766 348380 187188 673145 599993 245762 039418 297443 152366 564434 893898 736831 214715 416806 232872 022448 293505 302133 028019 601251 945155 409480 948326 635115 281591 072223 247162 524928 665364 063136 726796 052317 989609 380981 192143 496606 597166 144050 556039 640664 921110 849559 693395 823329 241437 034126 527365 986192 261841 019857 411797 945835 773910 771671 498428 568696 246986 344726 699648 038894 674313 302947 148221 142534 803155 608721 802273 523423 393179 032395 603559 632942 754816 002096 434228 989604 243165 857929 585357 836470 125580 727838 780546 046783 152878 981911 009533 852750 410621 170252 089565 316195 666490 020426 850859 545768 577589 419183 727222 633851 < 271267 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2798, 1365, F27, 84) (dual of [1365, 1267, 85]-code), but
(99−85, 99, 1366)-Net in Base 27 — Upper bound on s
There is no (14, 99, 1367)-net in base 27, because
- 1 times m-reduction [i] would yield (14, 98, 1367)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 188 449793 052710 695151 597139 180240 688042 618811 397807 821094 815803 578746 619730 985859 671086 007482 492910 593039 402826 924806 706991 387230 163308 045549 > 2798 [i]