MinT - Best Known (146−71, 146, s)-Nets in Base 5

Best Known (146−71, 146, s)-Nets in Base 5

(146−71, 146, 98)-Net over F₅ — Constructive and digital

Digital (75, 146, 98)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (9, 44, 26)-net over F₅, using
  - net from sequence [i] based on digital (9, 25)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 9 and N(F) ≥ 26, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 102, 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]

(146−71, 146, 156)-Net over F₅ — Digital

Digital (75, 146, 156)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(146−71, 146, 2709)-Net in Base 5 — Upper bound on s

There is no (75, 146, 2710)-net in base 5, because

1 times m-reduction [i] would yield (75, 145, 2710)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 226496 946388 042845 410358 584302 343792 663030 397155 896456 675585 872751 830596 212451 525957 290646 406405 054585 > 5¹⁴⁵ [i]