Best Known (131, ∞, s)-Nets in Base 3
(131, ∞, 78)-Net over F3 — Constructive and digital
Digital (131, m, 78)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (131, 77)-sequence over F3, using
- t-expansion [i] based on digital (121, 77)-sequence over F3, using
- base reduction for sequences [i] based on digital (22, 77)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 22 and N(F) ≥ 78, using
- F3 from the tower of function fields by GarcÃa and Stichtenoth over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 22 and N(F) ≥ 78, using
- base reduction for sequences [i] based on digital (22, 77)-sequence over F9, using
- t-expansion [i] based on digital (121, 77)-sequence over F3, using
(131, ∞, 120)-Net over F3 — Digital
Digital (131, m, 120)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (131, 119)-sequence over F3, using
- t-expansion [i] based on digital (113, 119)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 113 and N(F) ≥ 120, using
- t-expansion [i] based on digital (113, 119)-sequence over F3, using
(131, ∞, 275)-Net in Base 3 — Upper bound on s
There is no (131, m, 276)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (131, 1649, 276)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31649, 276, S3, 6, 1518), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 10511 332359 330779 444458 397472 168292 716795 712990 424003 895929 287004 853942 857445 329796 888837 282975 784367 173223 772885 711881 607985 303443 281987 990262 254815 476860 331511 397121 087106 210855 868781 910074 745824 908139 416659 510513 432316 569552 598566 006960 542534 847471 976150 913271 261555 318335 294655 446409 260356 727673 342872 531330 655624 877479 667880 384724 636065 858640 338841 212312 560777 834353 210122 779300 507303 406202 442193 478368 250632 941598 640426 826074 637366 597153 448913 923340 956893 642604 468805 971097 909920 533184 987063 852377 380076 880048 347931 712610 030462 515833 307389 373400 850085 098588 854925 314802 580386 010121 276542 965672 843393 054641 379845 376180 490115 260585 832584 333364 391766 806309 770240 462143 020558 447449 254958 549445 610205 728001 291920 909639 537904 027152 427435 277429 440019 474833 320170 315809 229214 736792 248603 185159 / 1519 > 31649 [i]
- extracting embedded OOA [i] would yield OOA(31649, 276, S3, 6, 1518), but