MinT - Best Known (150−105, 150, s)-Nets in Base 5

Best Known (150−105, 150, s)-Nets in Base 5

(150−105, 150, 78)-Net over F₅ — Constructive and digital

Digital (45, 150, 78)-net over F₅, using

t-expansion [i] based on digital (38, 150, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(150−105, 150, 88)-Net over F₅ — Digital

Digital (45, 150, 88)-net over F₅, using

net from sequence [i] based on digital (45, 87)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 45 and N(F) ≥ 88, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(150−105, 150, 471)-Net in Base 5 — Upper bound on s

There is no (45, 150, 472)-net in base 5, because

1 times m-reduction [i] would yield (45, 149, 472)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 148 627710 099117 499829 495303 539641 992813 057980 222953 928392 467022 681607 018385 864365 762695 812596 820435 329025 > 5¹⁴⁹ [i]