MinT - Best Known (90, 90+137, s)-Nets in Base 4

Best Known (90, 90+137, s)-Nets in Base 4

(90, 90+137, 104)-Net over F₄ — Constructive and digital

Digital (90, 227, 104)-net over F₄, using

t-expansion [i] based on digital (73, 227, 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]

(90, 90+137, 129)-Net over F₄ — Digital

Digital (90, 227, 129)-net over F₄, using

t-expansion [i] based on digital (81, 227, 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]

(90, 90+137, 818)-Net in Base 4 — Upper bound on s

There is no (90, 227, 819)-net in base 4, because

1 times m-reduction [i] would yield (90, 226, 819)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11666 349227 616061 052647 801094 885598 203967 695740 573360 717565 225148 196356 732243 390491 236442 674972 206687 374294 833274 202750 287164 898905 142032 > 4²²⁶ [i]