MinT - Best Known (43, 43+45, s)-Nets in Base 27

Best Known (43, 43+45, s)-Nets in Base 27

(43, 43+45, 190)-Net over F₂₇ — Constructive and digital

Digital (43, 88, 190)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 32, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]
2. digital (11, 56, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(43, 43+45, 370)-Net in Base 27 — Constructive

(43, 88, 370)-net in base 27, using

20 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(43, 43+45, 492)-Net over F₂₇ — Digital

Digital (43, 88, 492)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(43, 43+45, 159309)-Net in Base 27 — Upper bound on s

There is no (43, 88, 159310)-net in base 27, because

1 times m-reduction [i] would yield (43, 87, 159310)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 33781 142304 838190 091802 058060 932644 796510 106363 927748 591261 444764 964693 966583 982890 309822 596508 853229 385888 862725 734828 478053 > 27⁸⁷ [i]