MinT - Best Known (34, 34+84, s)-Nets in Base 4

Best Known (34, 34+84, s)-Nets in Base 4

(34, 34+84, 56)-Net over F₄ — Constructive and digital

Digital (34, 118, 56)-net over F₄, using

t-expansion [i] based on digital (33, 118, 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]

(34, 34+84, 65)-Net over F₄ — Digital

Digital (34, 118, 65)-net over F₄, using

t-expansion [i] based on digital (33, 118, 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]

(34, 34+84, 147)-Net in Base 4 — Upper bound on s

There is no (34, 118, 148)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4¹¹⁸, 148, S₄, 84), but
- the linear programming bound shows that MÂ â‰¥ 7301 270043 157677 378834 968797 352546 115167 116693 516048 447726 743341 396059 833062 672218 026684 710912 / 58934 105814 768105 796875 > 4¹¹⁸ [i]