MinT - Best Known (45, 45+69, s)-Nets in Base 5

Best Known (45, 45+69, s)-Nets in Base 5

(45, 45+69, 78)-Net over F₅ — Constructive and digital

Digital (45, 114, 78)-net over F₅, using

t-expansion [i] based on digital (38, 114, 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]

(45, 45+69, 88)-Net over F₅ — Digital

Digital (45, 114, 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]

(45, 45+69, 687)-Net in Base 5 — Upper bound on s

There is no (45, 114, 688)-net in base 5, because

1 times m-reduction [i] would yield (45, 113, 688)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 9 966342 261458 944338 430533 929561 130700 936070 730445 925591 039549 678757 112483 230465 > 5¹¹³ [i]