MinT - Best Known (150−77, 150, s)-Nets in Base 5

Best Known (150−77, 150, s)-Nets in Base 5

(150−77, 150, 90)-Net over F₅ — Constructive and digital

Digital (73, 150, 90)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 42, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 108, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(150−77, 150, 133)-Net over F₅ — Digital

Digital (73, 150, 133)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(150−77, 150, 2039)-Net in Base 5 — Upper bound on s

There is no (73, 150, 2040)-net in base 5, because

1 times m-reduction [i] would yield (73, 149, 2040)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 141 125962 050092 917053 932611 897451 089073 739613 878021 459854 627003 142764 811439 637611 827111 179611 606814 964609 > 5¹⁴⁹ [i]