MinT - Best Known (90−56, 90, s)-Nets in Base 32

Best Known (90−56, 90, s)-Nets in Base 32

(90−56, 90, 128)-Net over F₃₂ — Constructive and digital

Digital (34, 90, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 31, 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, 59, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(90−56, 90, 257)-Net in Base 32 — Constructive

(34, 90, 257)-net in base 32, using

base change [i] based on (19, 75, 257)-net in base 64, using
- 1 times m-reduction [i] based on (19, 76, 257)-net in base 64, using
  - base change [i] based on digital (0, 57, 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]

(90−56, 90, 273)-Net over F₃₂ — Digital

Digital (34, 90, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 90, 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]

(90−56, 90, 315)-Net in Base 32

(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]

(90−56, 90, 25082)-Net in Base 32 — Upper bound on s

There is no (34, 90, 25083)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2907 982605 969043 230671 301452 563011 508184 020985 201388 937980 657232 325868 894823 837564 506382 991748 459570 365413 241151 219148 701862 457028 326040 > 32⁹⁰ [i]