MinT - Best Known (49−31, 49, s)-Nets in Base 4

Best Known (49−31, 49, s)-Nets in Base 4

(49−31, 49, 33)-Net over F₄ — Constructive and digital

Digital (18, 49, 33)-net over F₄, using

t-expansion [i] based on digital (15, 49, 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]

(49−31, 49, 41)-Net over F₄ — Digital

Digital (18, 49, 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]

(49−31, 49, 169)-Net in Base 4 — Upper bound on s

There is no (18, 49, 170)-net in base 4, because

1 times m-reduction [i] would yield (18, 48, 170)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 85250 176828 905333 890777 032432 > 4⁴⁸ [i]