MinT - Best Known (40−21, 40, s)-Nets in Base 4

Best Known (40−21, 40, s)-Nets in Base 4

(40−21, 40, 33)-Net over F₄ — Constructive and digital

Digital (19, 40, 33)-net over F₄, using

t-expansion [i] based on digital (15, 40, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(40−21, 40, 41)-Net over F₄ — Digital

Digital (19, 40, 41)-net over F₄, using

t-expansion [i] based on digital (18, 40, 41)-net over F₄, using
- net from sequence [i] based on digital (18, 40)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
    - Drinfeld modules of rank 1 [i]

(40−21, 40, 328)-Net in Base 4 — Upper bound on s

There is no (19, 40, 329)-net in base 4, because

1 times m-reduction [i] would yield (19, 39, 329)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 305271 532095 934403 008816 > 4³⁹ [i]