MinT - Best Known (51, 51+94, s)-Nets in Base 3

Best Known (51, 51+94, s)-Nets in Base 3

Digital (51, 145, 48)-net over F₃, using

Digital (51, 145, 64)-net over F₃, using

t-expansion [i] based on digital (49, 145, 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 (51, 145, 204)-net over F₃, because

1 times m-reduction [i] would yield digital (51, 144, 204)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁴, 204, F₃, 93) (dual of [204, 60, 94]-code), but
  - residual code [i] would yield OA(3⁵¹, 110, S₃, 31), but
    - the linear programming bound shows that MÂ â‰¥ 27103 508958 616660 099388 058735 738039 757568 992407 234162 611825 882462 948846 428970 990364 625160 807452 274719 095060 078872 027349 329955 381926 561202 960333 105446 138240 280517 341767 410662 123867 211211 555448 / 12088 872897 616793 226656 425391 668285 319370 292783 656670 426139 876036 131456 534838 272045 441251 954158 313476 909583 033037 899142 334642 440098 569170 724363 206755 577542 040480 373885 > 3⁵¹ [i]

There is no (51, 145, 229)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1568 877380 422642 602438 651764 783929 066932 242128 725266 730074 012673 060987 > 3¹⁴⁵ [i]