MinT - Best Known (75−39, 75, s)-Nets in Base 27

Best Known (75−39, 75, s)-Nets in Base 27

(75−39, 75, 176)-Net over F₂₇ — Constructive and digital

Digital (36, 75, 176)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 26, 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 (10, 49, 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]

(75−39, 75, 370)-Net in Base 27 — Constructive

(36, 75, 370)-net in base 27, using

5 times m-reduction [i] based on (36, 80, 370)-net in base 27, using
- base change [i] based on digital (16, 60, 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]

(75−39, 75, 395)-Net over F₂₇ — Digital

Digital (36, 75, 395)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−39, 75, 114548)-Net in Base 27 — Upper bound on s

There is no (36, 75, 114549)-net in base 27, because

1 times m-reduction [i] would yield (36, 74, 114549)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 8335 557456 366229 350508 488887 184285 592420 146779 507677 680165 014935 720249 475845 985035 824160 450998 582729 599187 > 27⁷⁴ [i]