MinT - Best Known (81−41, 81, s)-Nets in Base 4

Best Known (81−41, 81, s)-Nets in Base 4

(81−41, 81, 56)-Net over F₄ — Constructive and digital

Digital (40, 81, 56)-net over F₄, using

t-expansion [i] based on digital (33, 81, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(81−41, 81, 75)-Net over F₄ — Digital

Digital (40, 81, 75)-net over F₄, using

net from sequence [i] based on digital (40, 74)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
  - Drinfeld modules of rank 1 [i]

(81−41, 81, 692)-Net in Base 4 — Upper bound on s

There is no (40, 81, 693)-net in base 4, because

1 times m-reduction [i] would yield (40, 80, 693)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 477545 371534 415598 042223 828739 201100 944477 668479 > 4⁸⁰ [i]