MinT - Best Known (49, 49+41, s)-Nets in Base 27

Best Known (49, 49+41, s)-Nets in Base 27

(49, 49+41, 216)-Net over F₂₇ — Constructive and digital

Digital (49, 90, 216)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 17, 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, 26, 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, 47, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(49, 49+41, 370)-Net in Base 27 — Constructive

(49, 90, 370)-net in base 27, using

t-expansion [i] based on (43, 90, 370)-net in base 27, using
- 18 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]

(49, 49+41, 1028)-Net over F₂₇ — Digital

Digital (49, 90, 1028)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 49+41, 747989)-Net in Base 27 — Upper bound on s

There is no (49, 90, 747990)-net in base 27, because

1 times m-reduction [i] would yield (49, 89, 747990)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 625067 283407 058923 744601 301384 279445 055396 810134 489572 010432 131460 923998 446015 161744 942228 597749 029595 184504 373881 480490 247849 > 27⁸⁹ [i]