MinT - Best Known (53, 53+97, s)-Nets in Base 3

Best Known (53, 53+97, s)-Nets in Base 3

Digital (53, 150, 48)-net over F₃, using

Digital (53, 150, 64)-net over F₃, using

t-expansion [i] based on digital (49, 150, 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, 150, 213)-net over F₃, because

1 times m-reduction [i] would yield digital (53, 149, 213)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁹, 213, F₃, 96) (dual of [213, 64, 97]-code), but
  - residual code [i] would yield OA(3⁵³, 116, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 30 210723 713137 067177 793975 595437 389937 655484 598142 095463 860309 406988 954987 805424 205733 738518 195687 105322 605711 836142 015912 625489 745728 349344 380080 374822 240851 452346 970066 591441 / 1 515367 982730 495420 433932 584698 701039 350523 187407 082539 706877 207417 526512 266220 467507 257976 353777 430603 205351 330224 160963 457142 806942 300281 987731 316143 > 3⁵³ [i]

There is no (53, 150, 240)-net in base 3, because

1 times m-reduction [i] would yield (53, 149, 240)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 140898 984481 663308 207271 463730 998831 164346 234581 056415 665136 067766 412289 > 3¹⁴⁹ [i]