MinT - Best Known (146−75, 146, s)-Nets in Base 5

Best Known (146−75, 146, s)-Nets in Base 5

(146−75, 146, 88)-Net over F₅ — Constructive and digital

Digital (71, 146, 88)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 40, 16)-net over F₅, using
  - net from sequence [i] based on digital (3, 15)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 106, 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]

(146−75, 146, 132)-Net over F₅ — Digital

Digital (71, 146, 132)-net over F₅, using

t-expansion [i] based on digital (67, 146, 132)-net over F₅, using
- net from sequence [i] based on digital (67, 131)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 67 and N(F) ≥ 132, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(146−75, 146, 1982)-Net in Base 5 — Upper bound on s

There is no (71, 146, 1983)-net in base 5, because

1 times m-reduction [i] would yield (71, 145, 1983)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 227346 464864 861510 537077 557089 613873 727671 748224 082335 219451 127061 055743 881379 416897 169153 237233 716845 > 5¹⁴⁵ [i]