MinT - Best Known (69−28, 69, s)-Nets in Base 32

Best Known (69−28, 69, s)-Nets in Base 32

(69−28, 69, 262)-Net over F₃₂ — Constructive and digital

Digital (41, 69, 262)-net over F₃₂, using

1 times m-reduction [i] based on digital (41, 70, 262)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 13, 66)-net over F₃₂, using
    - (u,Â u+v)-construction [i] based on
      1. digital (0, 4, 33)-net over F₃₂, using
        net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
        Niederreiter sequence [i]
      2. digital (0, 9, 33)-net over F₃₂, using
        net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  2. digital (7, 21, 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]
  3. digital (7, 36, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(69−28, 69, 514)-Net in Base 32 — Constructive

(41, 69, 514)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (9, 23, 257)-net in base 32, using
  - 1 times m-reduction [i] based on (9, 24, 257)-net in base 32, using
    - base change [i] based on digital (0, 15, 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]
2. (18, 46, 257)-net in base 32, using
  - 2 times m-reduction [i] based on (18, 48, 257)-net in base 32, using
    - base change [i] based on digital (0, 30, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(69−28, 69, 2488)-Net over F₃₂ — Digital

Digital (41, 69, 2488)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−28, 69, 5108985)-Net in Base 32 — Upper bound on s

There is no (41, 69, 5108986)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 71 672014 427424 607213 311388 585919 293429 701592 899573 399029 533308 391150 325951 567196 479713 829901 321359 267048 > 32⁶⁹ [i]