MinT - Best Known (233−155, 233, s)-Nets in Base 3

Best Known (233−155, 233, s)-Nets in Base 3

(233−155, 233, 53)-Net over F₃ — Constructive and digital

Digital (78, 233, 53)-net over F₃, using

net from sequence [i] based on digital (78, 52)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 52)-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]

(233−155, 233, 84)-Net over F₃ — Digital

Digital (78, 233, 84)-net over F₃, using

t-expansion [i] based on digital (71, 233, 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]

(233−155, 233, 246)-Net over F₃ — Upper bound on s (digital)

There is no digital (78, 233, 247)-net over F₃, because

2 times m-reduction [i] would yield digital (78, 231, 247)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³¹, 247, F₃, 153) (dual of [247, 16, 154]-code), but
  - residual code [i] would yield OA(3⁷⁸, 93, S₃, 51), but
    - the linear programming bound shows that MÂ â‰¥ 2117 985825 855951 548682 979121 686352 501183 391624 / 117 447583 > 3⁷⁸ [i]

(233−155, 233, 332)-Net in Base 3 — Upper bound on s

There is no (78, 233, 333)-net in base 3, because

1 times m-reduction [i] would yield (78, 232, 333)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 513 821562 910027 690212 506904 663021 209313 493412 849343 654456 142106 230490 093818 841298 648467 707710 048038 592286 802955 > 3²³² [i]