MinT - Best Known (81−35, 81, s)-Nets in Base 27

Best Known (81−35, 81, s)-Nets in Base 27

(81−35, 81, 228)-Net over F₂₇ — Constructive and digital

Digital (46, 81, 228)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 17, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (6, 23, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (6, 41, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(81−35, 81, 370)-Net in Base 27 — Constructive

(46, 81, 370)-net in base 27, using

t-expansion [i] based on (43, 81, 370)-net in base 27, using
- 27 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]

(81−35, 81, 1355)-Net over F₂₇ — Digital

Digital (46, 81, 1355)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−35, 81, 1502445)-Net in Base 27 — Upper bound on s

There is no (46, 81, 1502446)-net in base 27, because

1 times m-reduction [i] would yield (46, 80, 1502446)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 229250 576728 617620 722572 045407 758930 049756 074245 617929 070439 533831 678509 362368 564341 739436 604330 495213 115771 918349 > 27⁸⁰ [i]