MinT - Best Known (94−39, 94, s)-Nets in Base 27

Best Known (94−39, 94, s)-Nets in Base 27

(94−39, 94, 252)-Net over F₂₇ — Constructive and digital

Digital (55, 94, 252)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 20, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (7, 26, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (9, 48, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(94−39, 94, 408)-Net in Base 27 — Constructive

(55, 94, 408)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. (35, 74, 370)-net in base 27, using
  - 2 times m-reduction [i] based on (35, 76, 370)-net in base 27, using
    - base change [i] based on digital (16, 57, 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]

(94−39, 94, 2026)-Net over F₂₇ — Digital

Digital (55, 94, 2026)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(94−39, 94, 3093069)-Net in Base 27 — Upper bound on s

There is no (55, 94, 3093070)-net in base 27, because

1 times m-reduction [i] would yield (55, 93, 3093070)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13 086700 869280 335077 843666 604805 546213 538770 742826 312484 577055 339206 424671 313102 448802 585369 978533 958436 921122 927889 723762 325264 773873 > 27⁹³ [i]