MinT - Best Known (239−153, 239, s)-Nets in Base 3

Best Known (239−153, 239, s)-Nets in Base 3

(239−153, 239, 61)-Net over F₃ — Constructive and digital

Digital (86, 239, 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]

(239−153, 239, 84)-Net over F₃ — Digital

Digital (86, 239, 84)-net over F₃, using

t-expansion [i] based on digital (71, 239, 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]

(239−153, 239, 374)-Net over F₃ — Upper bound on s (digital)

There is no digital (86, 239, 375)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³⁹, 375, F₃, 153) (dual of [375, 136, 154]-code), but
- residual code [i] would yield linear OA(3⁸⁶, 221, F₃, 51) (dual of [221, 135, 52]-code), but
  - 1 times truncation [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]

(239−153, 239, 383)-Net in Base 3 — Upper bound on s

There is no (86, 239, 384)-net in base 3, because

1 times m-reduction [i] would yield (86, 238, 384)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 389583 540596 915367 159210 923058 100529 051376 710839 033816 467242 243411 902142 034128 438921 612049 933729 051079 411821 151233 > 3²³⁸ [i]