MinT - Best Known (77, 77+93, s)-Nets in Base 3

Best Known (77, 77+93, s)-Nets in Base 3

(77, 77+93, 56)-Net over F₃ — Constructive and digital

Digital (77, 170, 56)-net over F₃, using

1 times m-reduction [i] based on digital (77, 171, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 62, 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, 109, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(77, 77+93, 84)-Net over F₃ — Digital

Digital (77, 170, 84)-net over F₃, using

t-expansion [i] based on digital (71, 170, 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]

(77, 77+93, 465)-Net in Base 3 — Upper bound on s

There is no (77, 170, 466)-net in base 3, because

1 times m-reduction [i] would yield (77, 169, 466)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 447 245927 799637 296747 109458 297138 627877 871316 692847 590395 865378 207266 790088 779181 > 3¹⁶⁹ [i]