MinT - Best Known (84, 84+121, s)-Nets in Base 4

Best Known (84, 84+121, s)-Nets in Base 4

(84, 84+121, 104)-Net over F₄ — Constructive and digital

Digital (84, 205, 104)-net over F₄, using

t-expansion [i] based on digital (73, 205, 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]

(84, 84+121, 129)-Net over F₄ — Digital

Digital (84, 205, 129)-net over F₄, using

t-expansion [i] based on digital (81, 205, 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]

(84, 84+121, 812)-Net in Base 4 — Upper bound on s

There is no (84, 205, 813)-net in base 4, because

1 times m-reduction [i] would yield (84, 204, 813)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 661 845938 351848 534909 147458 520620 702350 722780 514892 411475 138312 775126 911113 124826 011822 595961 198060 525458 715428 882155 559824 > 4²⁰⁴ [i]