MinT - Best Known (62, 62+45, s)-Nets in Base 27

Best Known (62, 62+45, s)-Nets in Base 27

(62, 62+45, 260)-Net over F₂₇ — Constructive and digital

Digital (62, 107, 260)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 22, 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 (8, 30, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]
3. digital (10, 55, 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]

(62, 62+45, 422)-Net in Base 27 — Constructive

(62, 107, 422)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (3, 25, 52)-net over F₂₇, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by SÃ©mirat [i]
2. (37, 82, 370)-net in base 27, using
  - 2 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
    - base change [i] based on digital (16, 63, 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+45, 2030)-Net over F₂₇ — Digital

Digital (62, 107, 2030)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(62, 62+45, 2744408)-Net in Base 27 — Upper bound on s

There is no (62, 107, 2744409)-net in base 27, because

1 times m-reduction [i] would yield (62, 106, 2744409)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 53 034799 465626 781466 855804 094385 312845 188864 647304 591790 154509 772209 828389 789268 277679 648707 449981 612941 949096 456575 623596 906263 701348 989740 442089 121657 > 27¹⁰⁶ [i]