MinT - Best Known (51, 51+97, s)-Nets in Base 3

Best Known (51, 51+97, s)-Nets in Base 3

Digital (51, 148, 48)-net over F₃, using

Digital (51, 148, 64)-net over F₃, using

t-expansion [i] based on digital (49, 148, 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, 148, 188)-net over F₃, because

1 times m-reduction [i] would yield digital (51, 147, 188)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁷, 188, F₃, 96) (dual of [188, 41, 97]-code), but
  - residual code [i] would yield OA(3⁵¹, 91, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 3 796003 121606 659684 644081 485270 254802 809922 702083 145243 272087 278428 845740 570091 586803 340207 313946 354006 591960 625547 / 1 756668 470243 430645 991636 297238 233510 101788 678484 940922 765709 380449 381903 813470 038571 507115 > 3⁵¹ [i]

There is no (51, 148, 227)-net in base 3, because

1 times m-reduction [i] would yield (51, 147, 227)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14568 955706 033181 027131 505736 047702 326991 066991 924636 525984 351902 768801 > 3¹⁴⁷ [i]