Best Known (100−85, 100, s)-Nets in Base 25
(100−85, 100, 126)-Net over F25 — Constructive and digital
Digital (15, 100, 126)-net over F25, using
- t-expansion [i] based on digital (10, 100, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
(100−85, 100, 140)-Net over F25 — Digital
Digital (15, 100, 140)-net over F25, using
- net from sequence [i] based on digital (15, 139)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 15 and N(F) ≥ 140, using
(100−85, 100, 1334)-Net over F25 — Upper bound on s (digital)
There is no digital (15, 100, 1335)-net over F25, because
- 1 times m-reduction [i] would yield digital (15, 99, 1335)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2599, 1335, F25, 84) (dual of [1335, 1236, 85]-code), but
- the Johnson bound shows that N ≤ 713491 540300 484595 080821 165724 168440 765454 014600 392657 971609 880691 776199 766926 579615 137537 854917 743672 234197 095997 188960 802331 502048 230005 601921 647323 267082 791095 868341 325178 688158 048763 102866 325141 740898 108328 792208 663741 565967 015402 341905 032478 042632 708749 483541 919115 740980 301175 340508 286559 216002 157035 624490 219979 496066 028202 808531 504393 830181 304833 340446 460794 408574 889433 873477 788399 418674 535628 197032 862024 583095 620837 133361 955210 117887 242751 844096 946005 960699 081708 805802 193402 001146 668637 171352 381984 429285 787449 760679 484505 687362 276047 161260 328391 381670 441695 249791 777311 175464 960244 265520 699146 756724 267567 626060 330075 906190 981332 443288 023876 139199 085187 266883 496412 420887 125656 084271 723965 139258 789719 161233 112394 344125 887717 186097 349030 471081 208762 132006 623163 716873 995667 461034 489645 886692 480771 349238 692769 336343 344803 741664 952669 299340 644244 916171 429176 431293 477513 807058 276670 115232 271470 975504 848598 611820 708602 897444 644342 580515 731542 088082 733401 180547 408452 700813 188465 496897 878577 185213 724323 170413 693351 436960 831379 401675 011401 302163 711230 406367 235553 997157 652915 139439 563347 601489 473562 400671 703170 459851 478124 838919 524295 104709 013886 560485 043024 802023 961587 246132 586705 184773 783339 522652 719730 627507 811191 142452 826507 596686 532458 024154 141165 354360 568780 391227 804420 164213 615729 927586 578803 631841 938868 476528 779698 126744 046925 605889 121875 084635 144330 516668 247593 426623 744374 607656 595387 384159 375434 667642 287043 963198 149708 838347 039348 519078 491434 237416 941444 616593 831375 965935 769586 451468 646216 419182 881022 504150 225876 146695 218145 827495 840336 911359 878282 482596 767449 694036 762235 809355 744106 067386 708450 697064 840754 985625 585134 102468 472884 419360 856663 647541 067197 513802 498218 845630 514880 014870 049276 630773 < 251236 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2599, 1335, F25, 84) (dual of [1335, 1236, 85]-code), but
(100−85, 100, 1335)-Net in Base 25 — Upper bound on s
There is no (15, 100, 1336)-net in base 25, because
- 1 times m-reduction [i] would yield (15, 99, 1336)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 2 542435 262564 915505 711443 518757 613777 157599 954374 578918 087161 516933 033036 251001 353248 929049 120468 598917 374952 557100 755730 319835 209622 321025 > 2599 [i]