MinT - Best Known (76, 76+105, s)-Nets in Base 3

Best Known (76, 76+105, s)-Nets in Base 3

(76, 76+105, 51)-Net over F₃ — Constructive and digital

Digital (76, 181, 51)-net over F₃, using

net from sequence [i] based on digital (76, 50)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 50)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(76, 76+105, 84)-Net over F₃ — Digital

Digital (76, 181, 84)-net over F₃, using

t-expansion [i] based on digital (71, 181, 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]

(76, 76+105, 404)-Net in Base 3 — Upper bound on s

There is no (76, 181, 405)-net in base 3, because

1 times m-reduction [i] would yield (76, 180, 405)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 83 969344 132669 341484 402988 979238 832218 252975 680188 813774 866537 382296 753290 071884 711441 > 3¹⁸⁰ [i]