Best Known (10, 10+71, s)-Nets in Base 27
(10, 10+71, 94)-Net over F27 — Constructive and digital
Digital (10, 81, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
(10, 10+71, 99)-Net over F27 — Digital
Digital (10, 81, 99)-net over F27, using
- t-expansion [i] based on digital (9, 81, 99)-net over F27, using
- net from sequence [i] based on digital (9, 98)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 99, using
- net from sequence [i] based on digital (9, 98)-sequence over F27, using
(10, 10+71, 980)-Net over F27 — Upper bound on s (digital)
There is no digital (10, 81, 981)-net over F27, because
- 1 times m-reduction [i] would yield digital (10, 80, 981)-net over F27, but
- extracting embedded orthogonal array [i] would yield linear OA(2780, 981, F27, 70) (dual of [981, 901, 71]-code), but
- the Johnson bound shows that N ≤ 442386 089870 057310 761129 095657 170535 238490 027062 683694 071673 010546 775037 328596 129946 973565 084352 427659 199208 964537 333867 100663 365081 277563 984405 083094 868566 977745 958887 223336 680560 249679 039409 875787 083504 039689 108727 563882 452348 438929 628312 163863 210558 548358 973309 895112 036481 424230 190630 878572 848591 822659 255518 508089 745092 328230 255039 117867 615458 858350 807387 755014 357420 338472 953998 769239 770557 609230 442930 975867 048214 855834 886177 662592 311855 351412 476260 901283 653293 634557 901316 279953 132267 583960 793256 256301 915021 204158 012988 562864 214219 715086 689391 425007 383104 696450 106791 062584 918774 229851 504238 611576 419346 152677 595291 861942 915945 856178 977220 258313 850166 880299 419247 257447 970108 423897 854154 799049 476772 760584 089678 889958 467241 886743 764480 145528 231610 866483 363352 677629 271528 611861 479762 464425 642512 998068 178117 910127 288175 214455 699225 482298 486720 898057 745161 296494 647854 049410 911682 755015 891886 205090 688277 535309 441859 439991 768149 866374 271022 763780 677238 251546 709970 543885 098405 439246 494209 306516 477473 210015 701150 428087 016959 483670 728459 327564 853516 619961 884580 341755 630515 758188 135314 081615 009235 160404 532993 701049 616956 249972 452923 917669 623965 742049 808347 990515 636921 226721 722965 586379 979056 883519 570336 610025 868979 787465 445887 852006 467744 892037 039355 229006 681008 269954 667033 435677 < 27901 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2780, 981, F27, 70) (dual of [981, 901, 71]-code), but
(10, 10+71, 981)-Net in Base 27 — Upper bound on s
There is no (10, 81, 982)-net in base 27, because
- 1 times m-reduction [i] would yield (10, 80, 982)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 3 265605 212303 222549 177958 637111 589983 520186 970238 978857 877296 603331 227239 316246 475712 779741 595171 522074 361721 224433 > 2780 [i]