MinT - Best Known (85, 85+143, s)-Nets in Base 4

Best Known (85, 85+143, s)-Nets in Base 4

(85, 85+143, 104)-Net over F₄ — Constructive and digital

Digital (85, 228, 104)-net over F₄, using

t-expansion [i] based on digital (73, 228, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(85, 85+143, 129)-Net over F₄ — Digital

Digital (85, 228, 129)-net over F₄, using

t-expansion [i] based on digital (81, 228, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(85, 85+143, 707)-Net in Base 4 — Upper bound on s

There is no (85, 228, 708)-net in base 4, because

1 times m-reduction [i] would yield (85, 227, 708)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48590 019459 092941 088777 567287 338262 808251 856936 189479 835211 500000 381726 340137 941736 942905 677050 628667 808584 880851 091307 891903 441286 400800 > 4²²⁷ [i]