MinT - Best Known (100−55, 100, s)-Nets in Base 27

Best Known (100−55, 100, s)-Nets in Base 27

(100−55, 100, 178)-Net over F₂₇ — Constructive and digital

Digital (45, 100, 178)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 35, 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]
2. digital (10, 65, 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]

(100−55, 100, 353)-Net over F₂₇ — Digital

Digital (45, 100, 353)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(100−55, 100, 370)-Net in Base 27 — Constructive

(45, 100, 370)-net in base 27, using

t-expansion [i] based on (43, 100, 370)-net in base 27, using
- 8 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]

(100−55, 100, 74419)-Net in Base 27 — Upper bound on s

There is no (45, 100, 74420)-net in base 27, because

1 times m-reduction [i] would yield (45, 99, 74420)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 5071 427702 538934 826953 484705 420889 122330 453500 246691 583884 450493 124776 680227 277592 853948 508004 809729 210539 889401 017029 587291 463092 179868 169233 > 27⁹⁹ [i]