MinT - Best Known (33, 33+87, s)-Nets in Base 4

Best Known (33, 33+87, s)-Nets in Base 4

(33, 33+87, 56)-Net over F₄ — Constructive and digital

Digital (33, 120, 56)-net over F₄, using

net from sequence [i] based on digital (33, 55)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
  - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(33, 33+87, 65)-Net over F₄ — Digital

Digital (33, 120, 65)-net over F₄, using

net from sequence [i] based on digital (33, 64)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
  - Drinfeld modules of rank 1 [i]

(33, 33+87, 139)-Net in Base 4 — Upper bound on s

There is no (33, 120, 140)-net in base 4, because

1 times m-reduction [i] would yield (33, 119, 140)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹¹⁹, 140, S₄, 86), but
  - the linear programming bound shows that MÂ â‰¥ 45 122196 667887 806300 359088 099758 089092 443390 694382 406689 969289 605366 875274 513697 734656 / 68 890158 845145 > 4¹¹⁹ [i]