MinT - Best Known (42−27, 42, s)-Nets in Base 4

Best Known (42−27, 42, s)-Nets in Base 4

(42−27, 42, 33)-Net over F₄ — Constructive and digital

Digital (15, 42, 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]

(42−27, 42, 35)-Net over F₄ — Digital

Digital (15, 42, 35)-net over F₄, using

net from sequence [i] based on digital (15, 34)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 35, using
  - a curve from the manYPoints database [i]

(42−27, 42, 139)-Net in Base 4 — Upper bound on s

There is no (15, 42, 140)-net in base 4, because

1 times m-reduction [i] would yield (15, 41, 140)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 010855 070320 374258 180608 > 4⁴¹ [i]