Best Known (140−123, 140, s)-Nets in Base 9
(140−123, 140, 74)-Net over F9 — Constructive and digital
Digital (17, 140, 74)-net over F9, using
- net from sequence [i] based on digital (17, 73)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 17 and N(F) ≥ 74, using
(140−123, 140, 250)-Net over F9 — Upper bound on s (digital)
There is no digital (17, 140, 251)-net over F9, because
- 6 times m-reduction [i] would yield digital (17, 134, 251)-net over F9, but
- extracting embedded orthogonal array [i] would yield linear OA(9134, 251, F9, 117) (dual of [251, 117, 118]-code), but
- residual code [i] would yield OA(917, 133, S9, 13), but
- 1 times truncation [i] would yield OA(916, 132, S9, 12), but
- the linear programming bound shows that M ≥ 366759 481988 110308 514537 173543 / 191 274702 307783 > 916 [i]
- 1 times truncation [i] would yield OA(916, 132, S9, 12), but
- residual code [i] would yield OA(917, 133, S9, 13), but
- extracting embedded orthogonal array [i] would yield linear OA(9134, 251, F9, 117) (dual of [251, 117, 118]-code), but
(140−123, 140, 296)-Net in Base 9 — Upper bound on s
There is no (17, 140, 297)-net in base 9, because
- 18 times m-reduction [i] would yield (17, 122, 297)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(9122, 297, S9, 105), but
- 3 times code embedding in larger space [i] would yield OA(9125, 300, S9, 105), but
- the linear programming bound shows that M ≥ 250 964478 929263 859888 642236 976868 421595 068667 666049 978835 394763 291958 887073 847478 825983 372036 755381 794590 352549 951743 114034 215278 064807 115737 905334 742420 701883 414570 580928 211073 920904 034326 337717 908593 019651 250581 487823 453013 453116 588416 721038 204517 104905 729479 756602 756412 595355 391772 048348 453890 474162 330345 914716 494639 532650 851125 500241 210556 905198 778525 533442 632738 166662 773969 050518 719757 940236 252368 841338 293400 540725 907451 552208 059994 483378 881929 279967 878588 686560 260749 745359 036025 / 1089 055881 030189 146996 300641 464410 558649 822234 887868 929003 948311 644467 560364 069156 074329 812174 467384 405210 967247 185458 593695 301936 902374 726538 013563 458141 169949 043000 930990 084331 514756 318378 462886 148710 911466 174775 604693 471005 308569 250546 887496 403382 232357 897943 021382 300040 311532 461291 558644 124079 441930 519744 616983 469717 683314 902379 260742 881241 529572 682658 442303 552437 > 9125 [i]
- 3 times code embedding in larger space [i] would yield OA(9125, 300, S9, 105), but
- extracting embedded orthogonal array [i] would yield OA(9122, 297, S9, 105), but