MinT - Best Known (66, 66+119, s)-Nets in Base 3

Best Known (66, 66+119, s)-Nets in Base 3

Digital (66, 185, 48)-net over F₃, using

Digital (66, 185, 64)-net over F₃, using

t-expansion [i] based on digital (49, 185, 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]

There is no digital (66, 185, 266)-net over F₃, because

2 times m-reduction [i] would yield digital (66, 183, 266)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸³, 266, F₃, 117) (dual of [266, 83, 118]-code), but
  - residual code [i] would yield OA(3⁶⁶, 148, S₃, 39), but
    - the linear programming bound shows that MÂ â‰¥ 12601 938564 535135 135650 704084 012491 922357 294021 765537 541685 645839 226422 815808 943462 058299 571552 747099 509046 004028 133055 653131 287247 568473 006567 967081 409762 196702 490529 324098 740118 016092 247038 547359 649040 681487 642341 125002 274190 164327 183856 840151 147540 583967 171391 388109 925272 913352 964800 / 384 059324 319862 832498 618445 532744 571583 007585 206309 391529 282484 838516 443217 531358 149541 245870 985345 825263 049658 436169 025865 545718 943340 473250 969889 220421 359249 299552 438338 207068 772943 366528 348901 499961 632144 757700 140721 271228 293983 702658 240983 860982 727927 > 3⁶⁶ [i]

There is no (66, 185, 289)-net in base 3, because

10 times m-reduction [i] would yield (66, 175, 289)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁷⁵, 289, S₃, 109), but
  - 11 times code embedding in larger space [i] would yield OA(3¹⁸⁶, 300, S₃, 109), but
    - the linear programming bound shows that MÂ â‰¥ 744 646766 850603 526568 601268 797422 932026 535817 992617 489413 629925 481692 835741 945421 210090 969950 381274 252816 504020 353660 158683 457200 925516 493661 440389 831805 338854 508258 811020 990136 433923 762970 235494 388055 889637 880458 633459 997426 817105 857222 614437 226587 / 10512 885684 310613 723066 595609 767913 116130 406095 478720 857818 594175 294259 025889 504412 156455 324830 169263 105843 888569 812652 562285 873819 443701 107860 316189 843301 339995 > 3¹⁸⁶ [i]