MinT - Best Known (40, 139, s)-Nets in Base 5

Best Known (40, 139, s)-Nets in Base 5

(40, 139, 78)-Net over F₅ — Constructive and digital

Digital (40, 139, 78)-net over F₅, using

t-expansion [i] based on digital (38, 139, 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]

(40, 139, 409)-Net in Base 5 — Upper bound on s

There is no (40, 139, 410)-net in base 5, because

1 times m-reduction [i] would yield (40, 138, 410)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3 147560 580790 377675 723763 629668 322159 392603 120816 820742 687492 477822 226008 259537 700099 686274 770025 > 5¹³⁸ [i]