MinT - Best Known (58, 58+47, s)-Nets in Base 27

Best Known (58, 58+47, s)-Nets in Base 27

(58, 58+47, 240)-Net over F₂₇ — Constructive and digital

Digital (58, 105, 240)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 21, 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 (7, 30, 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]
3. digital (7, 54, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(58, 58+47, 370)-Net in Base 27 — Constructive

(58, 105, 370)-net in base 27, using

t-expansion [i] based on (43, 105, 370)-net in base 27, using
- 3 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]

(58, 58+47, 1304)-Net over F₂₇ — Digital

Digital (58, 105, 1304)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 58+47, 1075800)-Net in Base 27 — Upper bound on s

There is no (58, 105, 1075801)-net in base 27, because

1 times m-reduction [i] would yield (58, 104, 1075801)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72750 775300 748599 585464 024067 638702 955733 312955 457985 237095 285251 153628 158844 416625 753530 831307 310247 696893 862995 017741 791724 034319 025596 677297 306643 > 27¹⁰⁴ [i]