MinT - Best Known (237−149, 237, s)-Nets in Base 4

Best Known (237−149, 237, s)-Nets in Base 4

(237−149, 237, 104)-Net over F₄ — Constructive and digital

Digital (88, 237, 104)-net over F₄, using

t-expansion [i] based on digital (73, 237, 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]

(237−149, 237, 129)-Net over F₄ — Digital

Digital (88, 237, 129)-net over F₄, using

t-expansion [i] based on digital (81, 237, 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]

(237−149, 237, 727)-Net in Base 4 — Upper bound on s

There is no (88, 237, 728)-net in base 4, because

1 times m-reduction [i] would yield (88, 236, 728)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12790 375064 906165 890665 583791 338679 714188 958934 716067 822969 020551 061702 824842 496898 407712 820957 356626 688230 853031 746347 067353 859453 927538 563766 > 4²³⁶ [i]