MinT - Best Known (50, 50+39, s)-Nets in Base 32

Best Known (50, 50+39, s)-Nets in Base 32

(50, 50+39, 260)-Net over F₃₂ — Constructive and digital

Digital (50, 89, 260)-net over F₃₂, using

2 times m-reduction [i] based on digital (50, 91, 260)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (3, 16, 64)-net over F₃₂, using
    - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
        a function field by SÃ©mirat [i]
  2. digital (7, 27, 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, 48, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(50, 50+39, 513)-Net in Base 32 — Constructive

(50, 89, 513)-net in base 32, using

t-expansion [i] based on (46, 89, 513)-net in base 32, using
- 19 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(50, 50+39, 1643)-Net over F₃₂ — Digital

Digital (50, 89, 1643)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+39, 2393718)-Net in Base 32 — Upper bound on s

There is no (50, 89, 2393719)-net in base 32, because

1 times m-reduction [i] would yield (50, 88, 2393719)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 839231 939326 550318 423630 817921 679259 548986 091412 609101 783779 466226 653164 541922 743359 561638 243654 729636 862456 823710 895759 814320 018836 > 32⁸⁸ [i]