MinT - Best Known (86, 86+138, s)-Nets in Base 3

Best Known (86, 86+138, s)-Nets in Base 3

(86, 86+138, 61)-Net over F₃ — Constructive and digital

Digital (86, 224, 61)-net over F₃, using

net from sequence [i] based on digital (86, 60)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 60)-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]

(86, 86+138, 84)-Net over F₃ — Digital

Digital (86, 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]

(86, 86+138, 404)-Net in Base 3 — Upper bound on s

There is no (86, 224, 405)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 77580 567202 070630 148437 476275 202070 236885 877839 792989 397728 420538 139190 439722 139107 975431 737202 904347 944507 > 3²²⁴ [i]