MinT - Best Known (142−75, 142, s)-Nets in Base 3

Best Known (142−75, 142, s)-Nets in Base 3

(142−75, 142, 56)-Net over F₃ — Constructive and digital

Digital (67, 142, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 52, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 90, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(142−75, 142, 72)-Net over F₃ — Digital

Digital (67, 142, 72)-net over F₃, using

net from sequence [i] based on digital (67, 71)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(142−75, 142, 446)-Net in Base 3 — Upper bound on s

There is no (67, 142, 447)-net in base 3, because

1 times m-reduction [i] would yield (67, 141, 447)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 19 284632 919408 078809 630228 535821 063225 042063 211465 913795 306503 689527 > 3¹⁴¹ [i]