MinT - Best Known (85, 85+123, s)-Nets in Base 4

Best Known (85, 85+123, s)-Nets in Base 4

(85, 85+123, 104)-Net over F₄ — Constructive and digital

Digital (85, 208, 104)-net over F₄, using

t-expansion [i] based on digital (73, 208, 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+123, 129)-Net over F₄ — Digital

Digital (85, 208, 129)-net over F₄, using

t-expansion [i] based on digital (81, 208, 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+123, 818)-Net in Base 4 — Upper bound on s

There is no (85, 208, 819)-net in base 4, because

1 times m-reduction [i] would yield (85, 207, 819)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45318 137510 727299 847373 090915 351672 841277 577950 684596 173459 785661 074146 176815 815376 018113 537311 393876 938965 491083 861248 494720 > 4²⁰⁷ [i]