MinT - Best Known (103−43, 103, s)-Nets in Base 27

Best Known (103−43, 103, s)-Nets in Base 27

(103−43, 103, 260)-Net over F₂₇ — Constructive and digital

Digital (60, 103, 260)-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 (8, 29, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]
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]

(103−43, 103, 422)-Net in Base 27 — Constructive

(60, 103, 422)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 52)-net over F₂₇, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by SÃ©mirat [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]

(103−43, 103, 2077)-Net over F₂₇ — Digital

Digital (60, 103, 2077)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(103−43, 103, 2991272)-Net in Base 27 — Upper bound on s

There is no (60, 103, 2991273)-net in base 27, because

1 times m-reduction [i] would yield (60, 102, 2991273)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 794456 514961 844020 767323 076560 865291 636542 908293 205279 766111 206109 748644 908589 768467 659184 836429 178940 406087 660808 562716 262849 349861 258845 687099 > 27¹⁰² [i]