MinT - Best Known (86, 86+163, s)-Nets in Base 3

Best Known (86, 86+163, s)-Nets in Base 3

(86, 86+163, 61)-Net over F₃ — Constructive and digital

Digital (86, 249, 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+163, 84)-Net over F₃ — Digital

Digital (86, 249, 84)-net over F₃, using

t-expansion [i] based on digital (71, 249, 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+163, 280)-Net over F₃ — Upper bound on s (digital)

There is no digital (86, 249, 281)-net over F₃, because

1 times m-reduction [i] would yield digital (86, 248, 281)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁸, 281, F₃, 162) (dual of [281, 33, 163]-code), but
  - residual code [i] would yield OA(3⁸⁶, 118, S₃, 54), but
    - the linear programming bound shows that MÂ â‰¥ 45701 949395 937920 998380 846258 431667 611949 437892 667647 676231 147527 / 421027 592046 440957 546375 > 3⁸⁶ [i]

(86, 86+163, 372)-Net in Base 3 — Upper bound on s

There is no (86, 249, 373)-net in base 3, because

1 times m-reduction [i] would yield (86, 248, 373)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 23681 360971 334350 462555 543539 481849 595959 576087 434700 110470 319251 867686 373299 697912 966264 674830 469889 009461 236630 285227 > 3²⁴⁸ [i]