MinT - Best Known (51, 51+31, s)-Nets in Base 32

Best Known (51, 51+31, s)-Nets in Base 32

(51, 51+31, 316)-Net over F₃₂ — Constructive and digital

Digital (51, 82, 316)-net over F₃₂, using

1 times m-reduction [i] based on digital (51, 83, 316)-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, 23, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
  3. digital (11, 43, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(51, 51+31, 545)-Net in Base 32 — Constructive

(51, 82, 545)-net in base 32, using

32² times duplication [i] based on (49, 80, 545)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. (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. (25, 56, 288)-net in base 32, using
    - base change [i] based on digital (9, 40, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(51, 51+31, 5067)-Net over F₃₂ — Digital

Digital (51, 82, 5067)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+31, large)-Net in Base 32 — Upper bound on s

There is no (51, 82, large)-net in base 32, because

29 times m-reduction [i] would yield (51, 53, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]