MinT - Best Known (62, 62+43, s)-Nets in Base 27

Best Known (62, 62+43, s)-Nets in Base 27

(62, 62+43, 270)-Net over F₂₇ — Constructive and digital

Digital (62, 105, 270)-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 (10, 31, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]
3. digital (10, 53, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(62, 62+43, 438)-Net in Base 27 — Constructive

(62, 105, 438)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. 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]
2. (36, 79, 370)-net in base 27, using
  - 1 times m-reduction [i] based on (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]

(62, 62+43, 2427)-Net over F₂₇ — Digital

Digital (62, 105, 2427)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(62, 62+43, 4094272)-Net in Base 27 — Upper bound on s

There is no (62, 105, 4094273)-net in base 27, because

1 times m-reduction [i] would yield (62, 104, 4094273)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72750 036530 914009 442828 025852 843485 149345 424175 817147 541209 237166 682641 786649 089274 293073 157270 386826 150989 812596 106190 027037 029144 560442 788875 782699 > 27¹⁰⁴ [i]