MinT - Best Known (81−33, 81, s)-Nets in Base 32

Best Known (81−33, 81, s)-Nets in Base 32

(81−33, 81, 294)-Net over F₃₂ — Constructive and digital

Digital (48, 81, 294)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 18, 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]
2. digital (7, 23, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (7, 40, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(81−33, 81, 515)-Net in Base 32 — Constructive

(48, 81, 515)-net in base 32, using

32¹ times duplication [i] based on (47, 80, 515)-net in base 32, using
- base change [i] based on digital (17, 50, 515)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 16, 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. digital (1, 34, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(81−33, 81, 2679)-Net over F₃₂ — Digital

Digital (48, 81, 2679)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−33, 81, 7360824)-Net in Base 32 — Upper bound on s

There is no (48, 81, 7360825)-net in base 32, because

1 times m-reduction [i] would yield (48, 80, 7360825)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 582253 997129 278605 162378 491791 031714 092585 665241 048957 371765 804931 419877 281637 904512 134730 052962 020047 951652 238694 433561 > 32⁸⁰ [i]