MinT - Best Known (171−105, 171, s)-Nets in Base 3

Best Known (171−105, 171, s)-Nets in Base 3

(171−105, 171, 48)-Net over F₃ — Constructive and digital

Digital (66, 171, 48)-net over F₃, using

t-expansion [i] based on digital (45, 171, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(171−105, 171, 64)-Net over F₃ — Digital

Digital (66, 171, 64)-net over F₃, using

t-expansion [i] based on digital (49, 171, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(171−105, 171, 291)-Net in Base 3 — Upper bound on s

There is no (66, 171, 292)-net in base 3, because

1 times m-reduction [i] would yield (66, 170, 292)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁷⁰, 292, S₃, 104), but
  - 8 times code embedding in larger space [i] would yield OA(3¹⁷⁸, 300, S₃, 104), but
    - the linear programming bound shows that MÂ â‰¥ 6 902598 515546 512619 644272 804376 270507 125740 276667 985228 947156 996265 472926 840494 137362 953358 692450 883118 752955 679464 930589 657232 566105 161939 746335 015884 862352 336664 835909 958288 946854 232809 160076 289068 254844 697311 526136 409274 036089 517621 426269 806437 032514 753077 503346 697441 069928 486135 137289 063361 564187 402569 374655 198830 747847 693126 269756 023599 796339 122464 952085 398414 103280 500207 454131 600785 178096 842305 615487 757743 268384 426848 822059 222754 016803 018575 226593 163992 514421 069901 649090 492183 503576 657367 992102 142473 102590 008091 554351 675924 434677 643438 845301 164896 336558 302335 939798 383586 397300 866110 125073 / 512485 877933 544969 423012 238322 131327 548455 957409 704582 198949 521549 954519 600089 598638 999975 187525 552320 065533 866179 416537 329049 030978 273910 061450 273001 890047 653011 890873 778649 703276 512809 188831 563376 968632 661504 934571 600286 866202 384230 774344 683054 135086 797131 953447 098849 711166 618890 360107 840433 368855 990292 900626 061338 373841 686737 386854 925965 138129 608151 633599 455604 222688 393033 522677 605846 710288 067969 401696 536436 157851 008594 992224 729511 920056 433728 665816 331421 616401 962696 671723 963369 165231 867634 387579 647799 581585 > 3¹⁷⁸ [i]

Best Known (171−105, 171, s)-Nets in Base 3

(171−105, 171, 48)-Net over F3 — Constructive and digital

(171−105, 171, 64)-Net over F3 — Digital

(171−105, 171, 291)-Net in Base 3 — Upper bound on s

(171−105, 171, 48)-Net over F₃ — Constructive and digital

(171−105, 171, 64)-Net over F₃ — Digital