MinT - Best Known (53, 154, s)-Nets in Base 3

Best Known (53, 154, s)-Nets in Base 3

Digital (53, 154, 48)-net over F₃, using

Digital (53, 154, 64)-net over F₃, using

t-expansion [i] based on digital (49, 154, 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 (53, 154, 198)-net over F₃, because

2 times m-reduction [i] would yield digital (53, 152, 198)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵², 198, F₃, 99) (dual of [198, 46, 100]-code), but
  - residual code [i] would yield OA(3⁵³, 98, S₃, 33), but
    - the linear programming bound shows that MÂ â‰¥ 70405 986787 631825 456516 749181 938054 391496 023445 621407 247393 974860 100167 193867 925850 156399 787754 027088 012405 632448 512652 067196 819333 / 3435 057828 133240 896934 525314 441640 529877 282946 429987 863449 445479 386348 848486 854894 335014 985183 953870 063104 > 3⁵³ [i]

There is no (53, 154, 235)-net in base 3, because

1 times m-reduction [i] would yield (53, 153, 235)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 415075 276235 816390 215762 985803 340606 958004 818095 119051 064683 766379 311381 > 3¹⁵³ [i]