MinT - Best Known (65−25, 65, s)-Nets in Base 32

Best Known (65−25, 65, s)-Nets in Base 32

(65−25, 65, 294)-Net over F₃₂ — Constructive and digital

Digital (40, 65, 294)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 14, 98)-net over F₃₂, using
  - s-reduction based on digital (6, 14, 99)-net over F₃₂, using
    - generalized (u,Â u+v)-construction [i] based on
      1. digital (0, 2, 33)-net over F₃₂, using
        kÂ =Â 2 construction [i]
      2. 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]
      3. digital (0, 8, 33)-net over F₃₂, using
        net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
2. digital (7, 19, 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, 32, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(65−25, 65, 517)-Net in Base 32 — Constructive

(40, 65, 517)-net in base 32, using

32¹ times duplication [i] based on (39, 64, 517)-net in base 32, using
- base change [i] based on digital (15, 40, 517)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 13, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
    2. digital (2, 27, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(65−25, 65, 3783)-Net over F₃₂ — Digital

Digital (40, 65, 3783)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−25, 65, large)-Net in Base 32 — Upper bound on s

There is no (40, 65, large)-net in base 32, because

23 times m-reduction [i] would yield (40, 42, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]