MinT - Best Known (140−85, 140, s)-Nets in Base 5

Best Known (140−85, 140, s)-Nets in Base 5

(140−85, 140, 82)-Net over F₅ — Constructive and digital

Digital (55, 140, 82)-net over F₅, using

t-expansion [i] based on digital (48, 140, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(140−85, 140, 108)-Net over F₅ — Digital

Digital (55, 140, 108)-net over F₅, using

net from sequence [i] based on digital (55, 107)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 55 and N(F) ≥ 108, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(140−85, 140, 818)-Net in Base 5 — Upper bound on s

There is no (55, 140, 819)-net in base 5, because

1 times m-reduction [i] would yield (55, 139, 819)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 14 653376 137879 758545 935066 118586 734246 474653 271647 500220 125133 542274 638870 825185 951275 290202 412761 > 5¹³⁹ [i]