MinT - Best Known (93−38, 93, s)-Nets in Base 27

Best Known (93−38, 93, s)-Nets in Base 27

(93−38, 93, 258)-Net over F₂₇ — Constructive and digital

Digital (55, 93, 258)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 19, 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 (10, 48, 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]

(93−38, 93, 418)-Net in Base 27 — Constructive

(55, 93, 418)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (2, 21, 48)-net over F₂₇, using
  - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
      - a function field by GarcÃa and Quoos [i]
2. (34, 72, 370)-net in base 27, using
  - base change [i] based on digital (16, 54, 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]

(93−38, 93, 2250)-Net over F₂₇ — Digital

Digital (55, 93, 2250)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(93−38, 93, 3093069)-Net in Base 27 — Upper bound on s

There is no (55, 93, 3093070)-net in base 27, because

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]