MinT - Best Known (53, 96, s)-Nets in Base 27

Best Known (53, 96, s)-Nets in Base 27

(53, 96, 228)-Net over F₂₇ — Constructive and digital

Digital (53, 96, 228)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 20, 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, 27, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (6, 49, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(53, 96, 370)-Net in Base 27 — Constructive

(53, 96, 370)-net in base 27, using

t-expansion [i] based on (43, 96, 370)-net in base 27, using
- 12 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]

(53, 96, 1208)-Net over F₂₇ — Digital

Digital (53, 96, 1208)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 96, 997083)-Net in Base 27 — Upper bound on s

There is no (53, 96, 997084)-net in base 27, because

1 times m-reduction [i] would yield (53, 95, 997084)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 9540 298082 247949 046902 847956 825746 819600 914490 608056 765498 836454 580688 350559 245547 339685 869990 175612 457660 510984 247007 366481 652208 912185 > 27⁹⁵ [i]