MinT - Best Known (51, 51+43, s)-Nets in Base 27

Best Known (51, 51+43, s)-Nets in Base 27

(51, 51+43, 216)-Net over F₂₇ — Constructive and digital

Digital (51, 94, 216)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 18, 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 (6, 27, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
3. digital (6, 49, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(51, 51+43, 370)-Net in Base 27 — Constructive

(51, 94, 370)-net in base 27, using

t-expansion [i] based on (43, 94, 370)-net in base 27, using
- 14 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 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]

(51, 51+43, 1036)-Net over F₂₇ — Digital

Digital (51, 94, 1036)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+43, 728466)-Net in Base 27 — Upper bound on s

There is no (51, 94, 728467)-net in base 27, because

1 times m-reduction [i] would yield (51, 93, 728467)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13 086974 041950 336719 877137 557951 174820 090677 717634 896607 605184 755383 733936 456055 559732 119971 442038 263198 591047 927385 858447 582899 437671 > 27⁹³ [i]