MinT - Best Known (55, 55+43, s)-Nets in Base 27

Best Known (55, 55+43, s)-Nets in Base 27

(55, 55+43, 240)-Net over F₂₇ — Constructive and digital

Digital (55, 98, 240)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 20, 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]
2. digital (7, 28, 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]
3. digital (7, 50, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(55, 55+43, 370)-Net in Base 27 — Constructive

(55, 98, 370)-net in base 27, using

t-expansion [i] based on (43, 98, 370)-net in base 27, using
- 10 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]

(55, 55+43, 1410)-Net over F₂₇ — Digital

Digital (55, 98, 1410)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 55+43, 1364750)-Net in Base 27 — Upper bound on s

There is no (55, 98, 1364751)-net in base 27, because

1 times m-reduction [i] would yield (55, 97, 1364751)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 954893 192458 409304 176178 987897 545176 875292 402816 189412 894342 424110 964674 747513 057319 488859 534251 096687 977688 913997 987293 975887 273254 313727 > 27⁹⁷ [i]