MinT - Best Known (27, 70, s)-Nets in Base 32

Best Known (27, 70, s)-Nets in Base 32

(27, 70, 128)-Net over F₃₂ — Constructive and digital

Digital (27, 70, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (3, 46, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(27, 70, 225)-Net over F₃₂ — Digital

Digital (27, 70, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 70, 225)-net over F₃₂, using
- net from sequence [i] based on digital (24, 224)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 24 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(27, 70, 257)-Net in Base 32 — Constructive

(27, 70, 257)-net in base 32, using

2 times m-reduction [i] based on (27, 72, 257)-net in base 32, using
- base change [i] based on digital (0, 45, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(27, 70, 262)-Net in Base 32

(27, 70, 262)-net in base 32, using

base change [i] based on digital (7, 50, 262)-net over F₁₂₈, using
- net from sequence [i] based on digital (7, 261)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 7 and N(F) ≥ 262, using
    - a curve from the manYPoints database [i]

(27, 70, 24685)-Net in Base 32 — Upper bound on s

There is no (27, 70, 24686)-net in base 32, because

1 times m-reduction [i] would yield (27, 69, 24686)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 71 723321 318852 452046 956640 644404 658993 969150 873329 132153 375572 577650 878521 925773 290688 363544 958100 587518 > 32⁶⁹ [i]