MinT - Best Known (237−167, 237, s)-Nets in Base 4

Best Known (237−167, 237, s)-Nets in Base 4

(237−167, 237, 66)-Net over F₄ — Constructive and digital

Digital (70, 237, 66)-net over F₄, using

t-expansion [i] based on digital (49, 237, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(237−167, 237, 105)-Net over F₄ — Digital

Digital (70, 237, 105)-net over F₄, using

net from sequence [i] based on digital (70, 104)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
  - Drinfeld modules of rank 1 [i]

(237−167, 237, 478)-Net in Base 4 — Upper bound on s

There is no (70, 237, 479)-net in base 4, because

1 times m-reduction [i] would yield (70, 236, 479)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12509 279734 734940 498652 707564 790954 676027 544569 961999 637684 899858 407615 601102 171932 897523 677303 595535 997330 988692 778385 868254 317402 001397 714088 > 4²³⁶ [i]