Best Known (90−78, 90, s)-Nets in Base 27
(90−78, 90, 96)-Net over F27 — Constructive and digital
Digital (12, 90, 96)-net over F27, using
- t-expansion [i] based on digital (11, 90, 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
(90−78, 90, 109)-Net over F27 — Digital
Digital (12, 90, 109)-net over F27, using
- net from sequence [i] based on digital (12, 108)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 12 and N(F) ≥ 109, using
(90−78, 90, 1167)-Net over F27 — Upper bound on s (digital)
There is no digital (12, 90, 1168)-net over F27, because
- extracting embedded orthogonal array [i] would yield linear OA(2790, 1168, F27, 78) (dual of [1168, 1078, 79]-code), but
- the Johnson bound shows that N ≤ 9 988351 512089 257440 047505 720631 099586 643105 510474 634596 631308 579589 550363 079843 129649 468420 201058 600046 383909 185293 646586 427884 515175 246422 246515 688570 012934 544705 795033 977708 272440 251831 577193 664420 405814 911836 049617 274543 456606 455042 866922 583657 754191 458049 635381 566660 615595 883866 519737 834691 731314 829889 887375 392669 002500 076887 457970 300584 068833 407064 317556 515435 185659 860673 746628 690679 382250 081621 340576 887492 380309 823046 581631 563917 975066 426888 753922 682773 137304 572367 855973 657977 040914 819183 114505 597281 158581 344482 745306 633542 554668 457543 488520 062716 303378 635764 304270 277251 059555 507585 219820 138610 494335 623184 499650 357150 845128 862787 011412 306127 542868 177486 535226 405508 178186 083789 075248 905098 186698 694273 499083 804868 940704 733065 238660 260694 194527 594240 370912 154304 408254 531102 110986 778536 481386 796047 247676 192989 696881 660540 724942 863154 409720 622496 757633 741499 165161 206524 679699 443802 961352 149645 406398 492637 717120 310493 931729 305527 459056 755220 861620 724642 631263 443832 741494 405327 294860 353543 894217 021291 663972 641905 223249 382553 415229 868699 304832 509673 620348 034638 933769 301196 703915 721842 105368 170322 705562 595172 674821 838588 088093 210759 011933 596627 207373 559499 908442 903666 856162 093735 854346 508423 360143 146317 212269 839718 267197 483715 194638 423454 066037 953924 346100 195030 301048 048185 373306 349825 198295 384291 297260 816917 371838 077089 070910 542474 410080 517514 520304 043003 167893 863393 095339 699221 423675 952004 350153 529117 339895 692266 763184 999314 301168 133314 918880 939198 634136 781470 839473 771173 150051 434578 374392 739433 287264 333973 213165 321484 < 271078 [i]
(90−78, 90, 1169)-Net in Base 27 — Upper bound on s
There is no (12, 90, 1170)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 670 988837 546945 512886 757126 366775 253910 286195 866590 809798 932499 546485 182187 840649 692681 243227 846192 923866 155496 723375 954955 843737 > 2790 [i]