MinT - Best Known (103−44, 103, s)-Nets in Base 27

Best Known (103−44, 103, s)-Nets in Base 27

(103−44, 103, 252)-Net over F₂₇ — Constructive and digital

Digital (59, 103, 252)-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, 29, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (9, 53, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(103−44, 103, 408)-Net in Base 27 — Constructive

(59, 103, 408)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (1, 23, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. (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]

(103−44, 103, 1764)-Net over F₂₇ — Digital

Digital (59, 103, 1764)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(103−44, 103, 1750901)-Net in Base 27 — Upper bound on s

There is no (59, 103, 1750902)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2694 445669 634988 363391 271758 707321 319074 610644 792631 898673 893762 874777 660162 115283 335284 222276 579494 867846 502844 180385 181510 673906 287925 155697 386453 > 27¹⁰³ [i]