MinT - Best Known (59, 102, s)-Nets in Base 27

Best Known (59, 102, s)-Nets in Base 27

(59, 102, 258)-Net over F₂₇ — Constructive and digital

Digital (59, 102, 258)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 21, 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, 28, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (10, 53, 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]

(59, 102, 418)-Net in Base 27 — Constructive

(59, 102, 418)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (2, 23, 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. (36, 79, 370)-net in base 27, using
  - 1 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]

(59, 102, 1922)-Net over F₂₇ — Digital

Digital (59, 102, 1922)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 102, 2556792)-Net in Base 27 — Upper bound on s

There is no (59, 102, 2556793)-net in base 27, because

1 times m-reduction [i] would yield (59, 101, 2556793)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 696084 471742 685107 812419 764147 700297 114010 737108 429724 691350 724535 874890 345381 518984 260607 350474 465248 179956 603063 351759 210915 849131 310148 623963 > 27¹⁰¹ [i]