MinT - Best Known (82, 82+142, s)-Nets in Base 3

Best Known (82, 82+142, s)-Nets in Base 3

(82, 82+142, 57)-Net over F₃ — Constructive and digital

Digital (82, 224, 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+142, 84)-Net over F₃ — Digital

Digital (82, 224, 84)-net over F₃, using

t-expansion [i] based on digital (71, 224, 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+142, 370)-Net in Base 3 — Upper bound on s

There is no (82, 224, 371)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 84683 125571 768894 449821 082906 247281 490116 344608 744252 167133 298411 470624 052026 754153 223262 899657 315547 135043 > 3²²⁴ [i]