MinT - Best Known (240−153, 240, s)-Nets in Base 3

Best Known (240−153, 240, s)-Nets in Base 3

(240−153, 240, 62)-Net over F₃ — Constructive and digital

Digital (87, 240, 62)-net over F₃, using

net from sequence [i] based on digital (87, 61)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 61)-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]

(240−153, 240, 84)-Net over F₃ — Digital

Digital (87, 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]

(240−153, 240, 383)-Net over F₃ — Upper bound on s (digital)

There is no digital (87, 240, 384)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁰, 384, F₃, 153) (dual of [384, 144, 154]-code), but
- residual code [i] would yield linear OA(3⁸⁷, 230, F₃, 51) (dual of [230, 143, 52]-code), but
  - 1 times truncation [i] would yield linear OA(3⁸⁶, 229, F₃, 50) (dual of [229, 143, 51]-code), but
    - the Johnson bound shows that NÂ â‰¤ 153 153734 778487 159186 543991 832667 435898 157821 534223 513753 575995 914628 < 3¹⁴³ [i]

(240−153, 240, 390)-Net in Base 3 — Upper bound on s

There is no (87, 240, 391)-net in base 3, because

1 times m-reduction [i] would yield (87, 239, 391)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 257988 307860 570981 997852 399964 213941 542495 580542 643008 826099 948684 849602 050739 667984 843991 486031 080495 721125 305961 > 3²³⁹ [i]