MinT - Best Known (85, 85+153, s)-Nets in Base 3

Best Known (85, 85+153, s)-Nets in Base 3

(85, 85+153, 60)-Net over F₃ — Constructive and digital

Digital (85, 238, 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+153, 84)-Net over F₃ — Digital

Digital (85, 238, 84)-net over F₃, using

t-expansion [i] based on digital (71, 238, 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+153, 365)-Net over F₃ — Upper bound on s (digital)

There is no digital (85, 238, 366)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³⁸, 366, F₃, 153) (dual of [366, 128, 154]-code), but
- residual code [i] would yield linear OA(3⁸⁵, 212, F₃, 51) (dual of [212, 127, 52]-code), but
  - 1 times truncation [i] would yield linear OA(3⁸⁴, 211, F₃, 50) (dual of [211, 127, 51]-code), but
    - the Johnson bound shows that NÂ â‰¤ 3 465392 580935 096296 712977 980638 603324 843243 174078 085715 358657 < 3¹²⁷ [i]

(85, 85+153, 377)-Net in Base 3 — Upper bound on s

There is no (85, 238, 378)-net in base 3, because

1 times m-reduction [i] would yield (85, 237, 378)-net in base 3, but
- 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]