MinT - Best Known (85, 85+152, s)-Nets in Base 3

Best Known (85, 85+152, s)-Nets in Base 3

(85, 85+152, 60)-Net over F₃ — Constructive and digital

Digital (85, 237, 60)-net over F₃, using

net from sequence [i] based on digital (85, 59)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 59)-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]

(85, 85+152, 84)-Net over F₃ — Digital

Digital (85, 237, 84)-net over F₃, using

t-expansion [i] based on digital (71, 237, 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]

(85, 85+152, 370)-Net over F₃ — Upper bound on s (digital)

There is no digital (85, 237, 371)-net over F₃, because

2 times m-reduction [i] would yield digital (85, 235, 371)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁵, 371, F₃, 150) (dual of [371, 136, 151]-code), but
  - residual code [i] would yield linear OA(3⁸⁵, 220, F₃, 50) (dual of [220, 135, 51]-code), but
    - the Johnson bound shows that NÂ â‰¤ 22905 993227 448745 319194 488151 012424 723463 351500 755461 393112 800141 < 3¹³⁵ [i]

(85, 85+152, 377)-Net in Base 3 — Upper bound on s

There is no (85, 237, 378)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 140538 533518 132760 072091 910225 178971 360502 755019 645630 092766 192167 296615 312912 873696 601229 276809 135211 803488 092409 > 3²³⁷ [i]