Best Known (109−97, 109, s)-Nets in Base 27
(109−97, 109, 96)-Net over F27 — Constructive and digital
Digital (12, 109, 96)-net over F27, using
- t-expansion [i] based on digital (11, 109, 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
(109−97, 109, 109)-Net over F27 — Digital
Digital (12, 109, 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
(109−97, 109, 1161)-Net over F27 — Upper bound on s (digital)
There is no digital (12, 109, 1162)-net over F27, because
- 13 times m-reduction [i] would yield digital (12, 96, 1162)-net over F27, but
- extracting embedded orthogonal array [i] would yield linear OA(2796, 1162, F27, 84) (dual of [1162, 1066, 85]-code), but
- the Johnson bound shows that N ≤ 66 989273 274357 757208 480939 577088 655846 050123 609752 164391 841744 456365 770019 480173 818100 838295 277376 472969 054378 456371 228457 583404 754690 307827 722488 066933 439670 089849 971885 555495 808790 450619 394415 175877 551691 659526 695378 607775 752173 930485 831207 680054 183878 449526 141639 275507 490580 277080 879200 545527 552696 830001 057916 934772 239941 590351 244469 880807 111238 221367 176980 982030 978608 267714 197062 067598 677071 181743 165749 129441 920305 476380 311148 783316 665964 343242 382378 843753 749579 215309 625463 112352 064433 179854 487253 210660 260481 538986 038590 020268 644748 978658 764715 061481 271046 158235 773365 603303 467316 926902 704781 573025 109870 864657 896486 827456 404160 248787 283502 749908 954671 510946 235452 266967 303556 254576 901357 538561 841478 567099 160488 940502 296115 633003 318957 135722 286812 278282 366482 117091 642769 301289 795739 206646 535860 172744 069844 751323 165353 182307 304077 120754 147078 878519 594875 667119 587504 740939 973854 779552 642552 102299 622265 775332 170824 645712 430767 751446 191900 757231 675482 201990 248595 254922 063459 001417 768346 106495 178535 158100 307390 451172 328159 750234 884008 200182 198201 599331 795676 905386 730418 004650 448560 055425 845242 392548 935717 647507 969629 712110 831928 028052 314028 142409 983817 331065 064764 155931 598117 409360 880487 658379 925474 870165 406791 939301 807172 282914 715860 087417 640622 072695 326253 878266 269267 724955 363367 740807 794279 573726 950450 931703 212737 130545 385619 727766 024002 761569 203931 204428 619667 164364 205335 576975 015576 391142 420384 858942 151026 574662 063471 066982 963393 570541 703147 004771 514115 263986 241304 932357 058225 012985 492814 678610 157365 < 271066 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2796, 1162, F27, 84) (dual of [1162, 1066, 85]-code), but
(109−97, 109, 1165)-Net in Base 27 — Upper bound on s
There is no (12, 109, 1166)-net in base 27, because
- 9 times m-reduction [i] would yield (12, 100, 1166)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 139227 860035 388664 012110 347842 662294 336272 284278 382607 203273 557552 549640 797403 335096 981798 089675 474724 518188 290787 516027 421861 168473 362271 137049 > 27100 [i]