MinT - Best Known (81, 81+143, s)-Nets in Base 4

Best Known (81, 81+143, s)-Nets in Base 4

(81, 81+143, 104)-Net over F₄ — Constructive and digital

Digital (81, 224, 104)-net over F₄, using

t-expansion [i] based on digital (73, 224, 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]

(81, 81+143, 129)-Net over F₄ — Digital

Digital (81, 224, 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]

(81, 81+143, 650)-Net in Base 4 — Upper bound on s

There is no (81, 224, 651)-net in base 4, because

1 times m-reduction [i] would yield (81, 223, 651)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 196 531824 896260 484856 181579 380767 512237 008762 920857 070600 112257 013009 132773 021347 560763 961154 080496 055546 368799 077812 504724 894460 838080 > 4²²³ [i]