MinT - Best Known (62, 62+47, s)-Nets in Base 27

Best Known (62, 62+47, s)-Nets in Base 27

(62, 62+47, 258)-Net over F₂₇ — Constructive and digital

Digital (62, 109, 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, 30, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (10, 57, 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]

(62, 62+47, 418)-Net in Base 27 — Constructive

(62, 109, 418)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (2, 25, 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, 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]

(62, 62+47, 1729)-Net over F₂₇ — Digital

Digital (62, 109, 1729)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(62, 62+47, 1908387)-Net in Base 27 — Upper bound on s

There is no (62, 109, 1908388)-net in base 27, because

1 times m-reduction [i] would yield (62, 108, 1908388)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38662 359275 237128 192544 199779 672262 021634 418575 227793 193042 287986 794953 481669 797433 693068 390519 755638 718291 867909 111867 662101 175625 420266 943021 240837 820497 > 27¹⁰⁸ [i]