MinT - Best Known (82, 82+109, s)-Nets in Base 3

Best Known (82, 82+109, s)-Nets in Base 3

(82, 82+109, 57)-Net over F₃ — Constructive and digital

Digital (82, 191, 57)-net over F₃, using

net from sequence [i] based on digital (82, 56)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 56)-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]

(82, 82+109, 84)-Net over F₃ — Digital

Digital (82, 191, 84)-net over F₃, using

t-expansion [i] based on digital (71, 191, 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]

(82, 82+109, 449)-Net in Base 3 — Upper bound on s

There is no (82, 191, 450)-net in base 3, because

1 times m-reduction [i] would yield (82, 190, 450)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 957570 450197 460428 576720 562278 358623 039652 241168 221366 129906 812447 576579 871061 971924 928157 > 3¹⁹⁰ [i]