MinT - Best Known (50, 50+35, s)-Nets in Base 32

Best Known (50, 50+35, s)-Nets in Base 32

(50, 50+35, 294)-Net over F₃₂ — Constructive and digital

Digital (50, 85, 294)-net over F₃₂, using

32¹ times duplication [i] based on digital (49, 84, 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, 24, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
  3. digital (7, 42, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(50, 50+35, 514)-Net in Base 32 — Constructive

(50, 85, 514)-net in base 32, using

32¹ times duplication [i] based on (49, 84, 514)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. (11, 28, 257)-net in base 32, using
    - base change [i] based on (3, 20, 257)-net in base 128, using
      - 4 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
        base change [i] based on digital (0, 21, 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. (21, 56, 257)-net in base 32, using
    - base change [i] based on digital (0, 35, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(50, 50+35, 2546)-Net over F₃₂ — Digital

Digital (50, 85, 2546)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+35, 6335699)-Net in Base 32 — Upper bound on s

There is no (50, 85, 6335700)-net in base 32, because

1 times m-reduction [i] would yield (50, 84, 6335700)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 707687 405677 138654 482605 481150 272381 306604 028656 409772 738944 562411 003297 659423 434597 599559 186085 360598 644286 380737 343121 227561 > 32⁸⁴ [i]