MinT - Best Known (85, 85+160, s)-Nets in Base 3

Best Known (85, 85+160, s)-Nets in Base 3

(85, 85+160, 60)-Net over F₃ — Constructive and digital

Digital (85, 245, 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+160, 84)-Net over F₃ — Digital

Digital (85, 245, 84)-net over F₃, using

t-expansion [i] based on digital (71, 245, 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+160, 282)-Net over F₃ — Upper bound on s (digital)

There is no digital (85, 245, 283)-net over F₃, because

1 times m-reduction [i] would yield digital (85, 244, 283)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁴, 283, F₃, 159) (dual of [283, 39, 160]-code), but
  - residual code [i] would yield OA(3⁸⁵, 123, S₃, 53), but
    - the linear programming bound shows that MÂ â‰¥ 18 556558 104235 515171 973211 626345 894318 538602 254870 483785 952441 / 483 199328 091023 596855 > 3⁸⁵ [i]

(85, 85+160, 368)-Net in Base 3 — Upper bound on s

There is no (85, 245, 369)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 884 172697 782973 020751 261389 712083 488760 067241 500910 877070 816617 312585 613921 631522 295064 882794 103324 399229 760997 955649 > 3²⁴⁵ [i]