MinT - Best Known (61, 61+45, s)-Nets in Base 27

Best Known (61, 61+45, s)-Nets in Base 27

(61, 61+45, 258)-Net over F₂₇ — Constructive and digital

Digital (61, 106, 258)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 22, 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 (7, 29, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (10, 55, 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]

(61, 61+45, 418)-Net in Base 27 — Constructive

(61, 106, 418)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (2, 24, 48)-net over F₂₇, using
  - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
      - a function field by GarcÃa and Quoos [i]
2. (37, 82, 370)-net in base 27, using
  - 2 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
    - base change [i] based on digital (16, 63, 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]

(61, 61+45, 1885)-Net over F₂₇ — Digital

Digital (61, 106, 1885)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(61, 61+45, 2362579)-Net in Base 27 — Upper bound on s

There is no (61, 106, 2362580)-net in base 27, because

1 times m-reduction [i] would yield (61, 105, 2362580)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 964248 625905 533867 649044 265959 075814 141659 256967 038395 282352 931553 878876 800166 226177 153814 971470 024462 533650 187479 204761 124691 242058 799960 211270 111161 > 27¹⁰⁵ [i]