MinT - Best Known (86−52, 86, s)-Nets in Base 32

Best Known (86−52, 86, s)-Nets in Base 32

(86−52, 86, 142)-Net over F₃₂ — Constructive and digital

Digital (34, 86, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 27, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]
2. digital (7, 59, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(86−52, 86, 258)-Net in Base 32 — Constructive

(34, 86, 258)-net in base 32, using

2 times m-reduction [i] based on (34, 88, 258)-net in base 32, using
- base change [i] based on digital (1, 55, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(86−52, 86, 273)-Net over F₃₂ — Digital

Digital (34, 86, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 86, 273)-net over F₃₂, using
- net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(86−52, 86, 315)-Net in Base 32

(34, 86, 315)-net in base 32, using

4 times m-reduction [i] based on (34, 90, 315)-net in base 32, using
- base change [i] based on digital (19, 75, 315)-net over F₆₄, using
  - net from sequence [i] based on digital (19, 314)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 19 and N(F) ≥ 315, using
      - a curve from the manYPoints database [i]

(86−52, 86, 32383)-Net in Base 32 — Upper bound on s

There is no (34, 86, 32384)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2773 494700 922415 357100 879946 725304 892973 236388 894477 499114 298047 424062 771059 672491 808308 459732 881767 676973 260911 775050 598048 648185 > 32⁸⁶ [i]