MinT - Best Known (30, 51, s)-Nets in Base 27

Best Known (30, 51, s)-Nets in Base 27

(30, 51, 196)-Net over F₂₇ — Constructive and digital

Digital (30, 51, 196)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 11, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 14, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (5, 26, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]

(30, 51, 370)-Net in Base 27 — Constructive

(30, 51, 370)-net in base 27, using

5 times m-reduction [i] based on (30, 56, 370)-net in base 27, using
- base change [i] based on digital (16, 42, 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]

(30, 51, 1437)-Net over F₂₇ — Digital

Digital (30, 51, 1437)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(30, 51, 2499314)-Net in Base 27 — Upper bound on s

There is no (30, 51, 2499315)-net in base 27, because

1 times m-reduction [i] would yield (30, 50, 2499315)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 369989 067266 678823 122628 285278 623705 842495 661650 438552 525524 426936 932293 > 27⁵⁰ [i]