MinT - Best Known (213−112, 213, s)-Nets in Base 3

Best Known (213−112, 213, s)-Nets in Base 3

(213−112, 213, 68)-Net over F₃ — Constructive and digital

Digital (101, 213, 68)-net over F₃, using

net from sequence [i] based on digital (101, 67)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 67)-sequence over F₉, using
  - s-reduction based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(213−112, 213, 96)-Net over F₃ — Digital

Digital (101, 213, 96)-net over F₃, using

t-expansion [i] based on digital (89, 213, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(213−112, 213, 654)-Net in Base 3 — Upper bound on s

There is no (101, 213, 655)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 430395 344666 675804 824558 418983 000643 296340 272297 080803 389615 898282 549301 393219 434469 846518 103246 714513 > 3²¹³ [i]