MinT - Best Known (240−156, 240, s)-Nets in Base 3

Best Known (240−156, 240, s)-Nets in Base 3

(240−156, 240, 59)-Net over F₃ — Constructive and digital

Digital (84, 240, 59)-net over F₃, using

net from sequence [i] based on digital (84, 58)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 58)-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]

(240−156, 240, 84)-Net over F₃ — Digital

Digital (84, 240, 84)-net over F₃, using

t-expansion [i] based on digital (71, 240, 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]

(240−156, 240, 283)-Net over F₃ — Upper bound on s (digital)

There is no digital (84, 240, 284)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁰, 284, F₃, 156) (dual of [284, 44, 157]-code), but
- residual code [i] would yield OA(3⁸⁴, 127, S₃, 52), but
  - the linear programming bound shows that MÂ â‰¥ 75093 992178 172944 979964 312324 767811 438315 782053 828257 111860 508367 900579 / 6 152332 044204 270968 974779 651985 > 3⁸⁴ [i]

(240−156, 240, 366)-Net in Base 3 — Upper bound on s

There is no (84, 240, 367)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 645535 369756 252241 800519 604335 548883 094570 795668 910177 730327 921326 332020 332963 875409 478199 469450 966545 368027 933381 > 3²⁴⁰ [i]