Best Known (16, 16+90, s)-Nets in Base 8
(16, 16+90, 65)-Net over F8 — Constructive and digital
Digital (16, 106, 65)-net over F8, using
- t-expansion [i] based on digital (14, 106, 65)-net over F8, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
(16, 16+90, 271)-Net over F8 — Upper bound on s (digital)
There is no digital (16, 106, 272)-net over F8, because
- extracting embedded orthogonal array [i] would yield linear OA(8106, 272, F8, 90) (dual of [272, 166, 91]-code), but
- construction Y1 [i] would yield
- OA(8105, 128, S8, 90), but
- the linear programming bound shows that M ≥ 21950 796382 814438 782744 864463 381336 127596 030073 856912 690696 782531 039738 810829 256630 497562 292754 935128 215485 677031 129088 / 296811 395949 608374 280875 > 8105 [i]
- linear OA(8166, 272, F8, 144) (dual of [272, 106, 145]-code), but
- discarding factors / shortening the dual code would yield linear OA(8166, 244, F8, 144) (dual of [244, 78, 145]-code), but
- residual code [i] would yield OA(822, 99, S8, 18), but
- the linear programming bound shows that M ≥ 7 461344 900772 163012 506367 600128 688128 / 94986 278747 950549 > 822 [i]
- residual code [i] would yield OA(822, 99, S8, 18), but
- discarding factors / shortening the dual code would yield linear OA(8166, 244, F8, 144) (dual of [244, 78, 145]-code), but
- OA(8105, 128, S8, 90), but
- construction Y1 [i] would yield
(16, 16+90, 295)-Net in Base 8 — Upper bound on s
There is no (16, 106, 296)-net in base 8, because
- 4 times m-reduction [i] would yield (16, 102, 296)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(8102, 296, S8, 86), but
- 4 times code embedding in larger space [i] would yield OA(8106, 300, S8, 86), but
- the linear programming bound shows that M ≥ 1 813677 875369 152272 942573 621196 635803 751018 569122 832333 904547 616932 539570 207271 570162 414270 186689 351457 296081 136706 550412 470642 195703 863438 843196 001370 665838 185358 641792 018172 364679 217484 628335 048624 343807 042717 307486 862460 898668 811055 605083 015058 362771 936907 178369 827768 336619 556421 972934 712366 195270 776938 420836 098170 782223 082681 191215 701147 981866 592385 175258 023020 676383 298813 952000 / 2 609523 090828 280360 399673 517605 921202 027302 398919 728637 474815 813018 035055 667380 402821 337345 325099 601925 819413 791707 710882 570618 298161 573338 933318 342533 235415 899785 945735 478446 574545 152392 020552 468875 081370 574983 328005 680478 033720 175186 826334 000457 530734 946272 661810 404405 406720 334371 309973 > 8106 [i]
- 4 times code embedding in larger space [i] would yield OA(8106, 300, S8, 86), but
- extracting embedded orthogonal array [i] would yield OA(8102, 296, S8, 86), but