MinT - Best Known (77−31, 77, s)-Nets in Base 32

Best Known (77−31, 77, s)-Nets in Base 32

(77−31, 77, 294)-Net over F₃₂ — Constructive and digital

Digital (46, 77, 294)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 17, 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, 22, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (7, 38, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(77−31, 77, 515)-Net in Base 32 — Constructive

(46, 77, 515)-net in base 32, using

base change [i] based on (24, 55, 515)-net in base 128, using
- 1 times m-reduction [i] based on (24, 56, 515)-net in base 128, using
  - base change [i] based on digital (17, 49, 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, 33, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(77−31, 77, 2851)-Net over F₃₂ — Digital

Digital (46, 77, 2851)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−31, 77, large)-Net in Base 32 — Upper bound on s

There is no (46, 77, large)-net in base 32, because

29 times m-reduction [i] would yield (46, 48, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]