MinT - Best Known (148−77, 148, s)-Nets in Base 3

Best Known (148−77, 148, s)-Nets in Base 3

(148−77, 148, 56)-Net over F₃ — Constructive and digital

Digital (71, 148, 56)-net over F₃, using

5 times m-reduction [i] based on digital (71, 153, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 56, 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, 97, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(148−77, 148, 84)-Net over F₃ — Digital

Digital (71, 148, 84)-net over F₃, using

net from sequence [i] based on digital (71, 83)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(148−77, 148, 490)-Net in Base 3 — Upper bound on s

There is no (71, 148, 491)-net in base 3, because

1 times m-reduction [i] would yield (71, 147, 491)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14549 509677 972357 837615 161426 435118 116546 477563 950702 408563 200472 067309 > 3¹⁴⁷ [i]