MinT - Best Known (93−39, 93, s)-Nets in Base 27

Best Known (93−39, 93, s)-Nets in Base 27

(93−39, 93, 248)-Net over F₂₇ — Constructive and digital

Digital (54, 93, 248)-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 (8, 47, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(93−39, 93, 398)-Net in Base 27 — Constructive

(54, 93, 398)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [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]

(93−39, 93, 1859)-Net over F₂₇ — Digital

Digital (54, 93, 1859)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(93−39, 93, 2600486)-Net in Base 27 — Upper bound on s

There is no (54, 93, 2600487)-net in base 27, because

1 times m-reduction [i] would yield (54, 92, 2600487)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 484695 425335 751934 186607 955738 346374 929789 912246 543246 480149 543552 803640 221799 420055 508682 367758 948510 606746 230169 059057 758061 806579 > 27⁹² [i]