MinT - Best Known (102−55, 102, s)-Nets in Base 27

Best Known (102−55, 102, s)-Nets in Base 27

(102−55, 102, 188)-Net over F₂₇ — Constructive and digital

Digital (47, 102, 188)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 37, 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]
2. digital (10, 65, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(102−55, 102, 370)-Net in Base 27 — Constructive

(47, 102, 370)-net in base 27, using

t-expansion [i] based on (43, 102, 370)-net in base 27, using
- 6 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]

(102−55, 102, 401)-Net over F₂₇ — Digital

Digital (47, 102, 401)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(102−55, 102, 95001)-Net in Base 27 — Upper bound on s

There is no (47, 102, 95002)-net in base 27, because

1 times m-reduction [i] would yield (47, 101, 95002)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 696701 355195 260235 658822 068964 647012 336714 309170 423679 764505 279059 373534 028587 438186 157431 414647 756195 934532 144512 830787 962545 218058 963201 876081 > 27¹⁰¹ [i]